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

Percentage Accurate: 83.3% → 98.0%

Time: 16.1s

Alternatives: 13

Speedup: 1.0×

• 27.2% 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: 83.3% 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: 98.0% accurate, 1.0× speedup

Math

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

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 (+ z 1.0))) 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 / (z + 1.0))) / 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 * (((x_m / z) * (y_m / (z + 1.0d0))) / 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 * (((x_m / z) * (y_m / (z + 1.0))) / 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 * (((x_m / z) * (y_m / (z + 1.0))) / 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(x_m / z) * Float64(y_m / Float64(z + 1.0))) / 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 * (((x_m / z) * (y_m / (z + 1.0))) / 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[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. *-commutative 85.5%

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

   2. associate-/l* 85.4%

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

   3. sqr-neg 85.4%

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

   4. associate-/r* 86.5%

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

   5. sqr-neg 86.5%

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

3. Simplified 86.5%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 89.3%

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

   2. *-commutative 89.3%

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

   3. associate-*r/ 88.9%

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

   4. associate-/r* 93.7%

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

   5. associate-*l/ 96.5%

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

6. Applied egg-rr 96.5%

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

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z + 1}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= (/ (* x_m y_m) (* (+ z 1.0) (* z z))) 2e-98)
       (* (/ y_m (* z z)) t_0)
       (* (/ y_m z) (/ t_0 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 t_0 = x_m / (z + 1.0);
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e-98) {
		tmp = (y_m / (z * z)) * t_0;
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	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 / (z + 1.0d0)
    if (((x_m * y_m) / ((z + 1.0d0) * (z * z))) <= 2d-98) then
        tmp = (y_m / (z * z)) * t_0
    else
        tmp = (y_m / z) * (t_0 / z)
    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 / (z + 1.0);
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e-98) {
		tmp = (y_m / (z * z)) * t_0;
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	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 / (z + 1.0)
	tmp = 0
	if ((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e-98:
		tmp = (y_m / (z * z)) * t_0
	else:
		tmp = (y_m / z) * (t_0 / 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)
	t_0 = Float64(x_m / Float64(z + 1.0))
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e-98)
		tmp = Float64(Float64(y_m / Float64(z * z)) * t_0);
	else
		tmp = Float64(Float64(y_m / z) * Float64(t_0 / z));
	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 / (z + 1.0);
	tmp = 0.0;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e-98)
		tmp = (y_m / (z * z)) * t_0;
	else
		tmp = (y_m / z) * (t_0 / z);
	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[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-98], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.99999999999999988e-98

   1. Initial program 88.8%

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

      1. *-commutative 88.8%

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

      2. sqr-neg 88.8%

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

      3. times-frac 93.1%

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

      4. sqr-neg 93.1%

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

   3. Simplified 93.1%

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

   if 1.99999999999999988e-98 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. frac-times 79.2%

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

      3. associate-*l/ 78.4%

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

      4. times-frac 97.1%

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

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.6× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
     (/ (/ (* x_m (/ y_m z)) z) z)
     (/ (/ y_m (/ 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m * (y_m / z)) / z) / z;
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x_m * (y_m / z)) / z) / z
    else
        tmp = (y_m / (z / x_m)) / z
    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 tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m * (y_m / z)) / z) / z;
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((x_m * (y_m / z)) / z) / z
	else:
		tmp = (y_m / (z / x_m)) / 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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x_m * Float64(y_m / z)) / z) / z);
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = ((x_m * (y_m / z)) / z) / z;
	else
		tmp = (y_m / (z / x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 88.0%

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

      3. sqr-neg 88.0%

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

      4. associate-/r* 89.9%

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

      5. sqr-neg 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 95.3%

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

      2. *-commutative 95.3%

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

      3. associate-*r/ 94.5%

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

      4. associate-/r* 96.7%

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

      5. associate-*l/ 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}} \]
   7. Step-by-step derivation

      1. frac-times 94.9%

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

      2. *-commutative 94.9%

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

      3. frac-times 98.8%

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

      4. associate-*r/ 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in z around inf 96.4%

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

   if -1 < z < 1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-/r* 82.6%

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

      5. sqr-neg 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r/ 82.5%

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

      4. associate-/r* 90.4%

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

      5. associate-*l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 88.6%

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

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*l/ 88.6%

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

       2. associate-*r/ 92.4%

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

       3. clear-num 92.4%

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

       4. un-div-inv 92.8%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 94.7%

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

Alternative 4: 96.4% accurate, 0.6× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
     (/ (* (/ x_m z) (/ y_m z)) z)
     (/ (/ y_m (/ 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x_m / z) * (y_m / z)) / z
    else
        tmp = (y_m / (z / x_m)) / z
    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 tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((x_m / z) * (y_m / z)) / z
	else:
		tmp = (y_m / (z / x_m)) / 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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = ((x_m / z) * (y_m / z)) / z;
	else
		tmp = (y_m / (z / x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 88.0%

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

      3. sqr-neg 88.0%

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

      4. associate-/r* 89.9%

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

      5. sqr-neg 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 95.3%

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

      2. *-commutative 95.3%

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

      3. associate-*r/ 94.5%

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

      4. associate-/r* 96.7%

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

      5. associate-*l/ 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in z around inf 96.9%

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

   if -1 < z < 1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-/r* 82.6%

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

      5. sqr-neg 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r/ 82.5%

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

      4. associate-/r* 90.4%

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

      5. associate-*l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 88.6%

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

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*l/ 88.6%

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

       2. associate-*r/ 92.4%

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

       3. clear-num 92.4%

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

       4. un-div-inv 92.8%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 95.0%

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

Alternative 5: 94.3% accurate, 0.6× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ x_m z) (/ y_m (* z z)))
     (/ (/ y_m (/ 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x_m / z) * (y_m / (z * z))
    else
        tmp = (y_m / (z / x_m)) / z
    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 tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (x_m / z) * (y_m / (z * z))
	else:
		tmp = (y_m / (z / x_m)) / 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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (x_m / z) * (y_m / (z * z));
	else
		tmp = (y_m / (z / x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. sqr-neg 90.1%

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

      3. times-frac 94.9%

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

      4. sqr-neg 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in z around inf 92.9%

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

   if -1 < z < 1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-/r* 82.6%

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

      5. sqr-neg 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r/ 82.5%

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

      4. associate-/r* 90.4%

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

      5. associate-*l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 88.6%

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

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*l/ 88.6%

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

       2. associate-*r/ 92.4%

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

       3. clear-num 92.4%

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

       4. un-div-inv 92.8%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 92.9%

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

Alternative 6: 94.9% accurate, 0.6× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ y_m z) (/ (/ x_m z) z))
     (/ (/ y_m (/ 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (y_m / z) * ((x_m / z) / z)
    else
        tmp = (y_m / (z / x_m)) / z
    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 tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (y_m / z) * ((x_m / z) / z)
	else:
		tmp = (y_m / (z / x_m)) / 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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / z) / z));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (y_m / z) * ((x_m / z) / z);
	else
		tmp = (y_m / (z / x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. frac-times 94.9%

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

      3. associate-*l/ 96.6%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around inf 94.8%

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

   if -1 < z < 1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-/r* 82.6%

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

      5. sqr-neg 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r/ 82.5%

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

      4. associate-/r* 90.4%

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

      5. associate-*l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 88.6%

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

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*l/ 88.6%

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

       2. associate-*r/ 92.4%

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

       3. clear-num 92.4%

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

       4. un-div-inv 92.8%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 93.8%

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

Alternative 7: 90.8% accurate, 0.6× speedup

Math

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

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 (or (<= z -1.0) (not (<= z 1.0)))
     (* y_m (/ (/ x_m (* z z)) z))
     (/ (/ y_m (/ 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * ((x_m / (z * z)) / z);
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = y_m * ((x_m / (z * z)) / z)
    else
        tmp = (y_m / (z / x_m)) / z
    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 tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * ((x_m / (z * z)) / z);
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	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):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = y_m * ((x_m / (z * z)) / z)
	else:
		tmp = (y_m / (z / x_m)) / 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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / z));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	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)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = y_m * ((x_m / (z * z)) / z);
	else
		tmp = (y_m / (z / x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 88.0%

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

      3. sqr-neg 88.0%

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

      4. associate-/r* 89.9%

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

      5. sqr-neg 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in z around inf 88.0%

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

   if -1 < z < 1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-/r* 82.6%

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

      5. sqr-neg 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r/ 82.5%

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

      4. associate-/r* 90.4%

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

      5. associate-*l/ 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 88.6%

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

      1. associate-*r/ 93.9%

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

      2. *-commutative 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*l/ 88.6%

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

       2. associate-*r/ 92.4%

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

       3. clear-num 92.4%

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

       4. un-div-inv 92.8%

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

   11. Applied egg-rr 92.8%

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

4. Final simplification 90.3%

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

Alternative 8: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 2e-115)
     (/ (/ x_m z) (/ z y_m))
     (* 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) {
	double tmp;
	if ((x_m * y_m) <= 2e-115) {
		tmp = (x_m / z) / (z / y_m);
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	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) :: tmp
    if ((x_m * y_m) <= 2d-115) then
        tmp = (x_m / z) / (z / y_m)
    else
        tmp = y_m * (x_m / (z * z))
    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 tmp;
	if ((x_m * y_m) <= 2e-115) {
		tmp = (x_m / z) / (z / y_m);
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	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):
	tmp = 0
	if (x_m * y_m) <= 2e-115:
		tmp = (x_m / z) / (z / y_m)
	else:
		tmp = y_m * (x_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(x_m * y_m) <= 2e-115)
		tmp = Float64(Float64(x_m / z) / Float64(z / y_m));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	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)
	tmp = 0.0;
	if ((x_m * y_m) <= 2e-115)
		tmp = (x_m / z) / (z / y_m);
	else
		tmp = y_m * (x_m / (z * z));
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-115], N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.0000000000000001e-115

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. frac-times 87.4%

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

      3. associate-*l/ 85.6%

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

      4. times-frac 95.2%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in z around 0 77.3%

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

      1. *-commutative 77.3%

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

      2. clear-num 77.3%

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

      3. un-div-inv 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 2.0000000000000001e-115 < (*.f64 x y)

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-/l* 89.4%

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

      3. sqr-neg 89.4%

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

      4. associate-/r* 90.7%

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

      5. sqr-neg 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in z around 0 69.9%

      \[\leadsto y \cdot \frac{\frac{x}{z \cdot z}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 1e+15)
     (/ (/ y_m (/ z x_m)) z)
     (* x_m (/ 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 ((x_m * y_m) <= 1e+15) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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) :: tmp
    if ((x_m * y_m) <= 1d+15) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = x_m * (y_m / (z * z))
    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 tmp;
	if ((x_m * y_m) <= 1e+15) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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):
	tmp = 0
	if (x_m * y_m) <= 1e+15:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = x_m * (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(x_m * y_m) <= 1e+15)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * z)));
	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)
	tmp = 0.0;
	if ((x_m * y_m) <= 1e+15)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = x_m * (y_m / (z * z));
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1e+15], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e15

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-/l* 84.9%

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

      3. sqr-neg 84.9%

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

      4. associate-/r* 85.8%

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

      5. sqr-neg 85.8%

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

   3. Simplified 85.8%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 88.2%

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

      2. *-commutative 88.2%

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

      3. associate-*r/ 87.7%

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

      4. associate-/r* 92.8%

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

      5. associate-*l/ 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in z around 0 73.1%

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

      1. associate-*r/ 76.8%

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

      2. *-commutative 76.8%

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

   9. Simplified 76.8%

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

       1. associate-*l/ 73.1%

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

       2. associate-*r/ 75.3%

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

       3. clear-num 75.4%

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

       4. un-div-inv 75.6%

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

   11. Applied egg-rr 75.6%

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

   if 1e15 < (*.f64 x y)

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

      2. sqr-neg 87.6%

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

      3. times-frac 94.9%

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

      4. sqr-neg 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in z around 0 71.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= x_m 2e-141) (/ (* (/ x_m z) y_m) z) (* x_m (/ 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 (x_m <= 2e-141) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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) :: tmp
    if (x_m <= 2d-141) then
        tmp = ((x_m / z) * y_m) / z
    else
        tmp = x_m * (y_m / (z * z))
    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 tmp;
	if (x_m <= 2e-141) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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):
	tmp = 0
	if x_m <= 2e-141:
		tmp = ((x_m / z) * y_m) / z
	else:
		tmp = x_m * (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 (x_m <= 2e-141)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * z)));
	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)
	tmp = 0.0;
	if (x_m <= 2e-141)
		tmp = ((x_m / z) * y_m) / z;
	else
		tmp = x_m * (y_m / (z * z));
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2e-141], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-141

   1. Initial program 82.6%

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

      1. *-commutative 82.6%

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

      2. associate-/l* 82.7%

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

      3. sqr-neg 82.7%

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

      4. associate-/r* 83.0%

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

      5. sqr-neg 83.0%

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

   3. Simplified 83.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*r/ 86.6%

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

      4. associate-/r* 93.6%

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

      5. associate-*l/ 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around 0 77.0%

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

   if 2.0000000000000001e-141 < x

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. sqr-neg 90.3%

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

      3. times-frac 92.5%

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

      4. sqr-neg 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around 0 71.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= x_m 2.8e-124) (* (/ x_m z) (/ y_m z)) (* x_m (/ 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 (x_m <= 2.8e-124) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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) :: tmp
    if (x_m <= 2.8d-124) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = x_m * (y_m / (z * z))
    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 tmp;
	if (x_m <= 2.8e-124) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	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):
	tmp = 0
	if x_m <= 2.8e-124:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = x_m * (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 (x_m <= 2.8e-124)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * z)));
	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)
	tmp = 0.0;
	if (x_m <= 2.8e-124)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = x_m * (y_m / (z * z));
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2.8e-124], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999998e-124

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. frac-times 87.9%

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

      3. associate-*l/ 85.4%

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

      4. times-frac 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in z around 0 77.7%

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

   if 2.79999999999999998e-124 < x

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. sqr-neg 90.2%

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

      3. times-frac 92.4%

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

      4. sqr-neg 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in z around 0 71.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.4% accurate, 1.0× speedup

Math

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

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) (/ (/ x_m (+ z 1.0)) 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 / (z + 1.0)) / 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) * ((x_m / (z + 1.0d0)) / 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) * ((x_m / (z + 1.0)) / 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) * ((x_m / (z + 1.0)) / 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(Float64(x_m / Float64(z + 1.0)) / 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) * ((x_m / (z + 1.0)) / 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[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. *-commutative 85.5%

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

   2. frac-times 89.6%

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

   3. associate-*l/ 88.9%

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

   4. times-frac 96.0%

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

4. Applied egg-rr 96.0%

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

Alternative 13: 74.9% accurate, 1.6× speedup

Math

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

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 (* (/ x_m z) (/ y_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 * ((x_m / z) * (y_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 * ((x_m / z) * (y_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 * ((x_m / z) * (y_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 * ((x_m / z) * (y_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(x_m / z) * Float64(y_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 * ((x_m / z) * (y_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[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. *-commutative 85.5%

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

   2. frac-times 89.6%

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

   3. associate-*l/ 88.9%

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

   4. times-frac 96.0%

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

4. Applied egg-rr 96.0%

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

5. Taylor expanded in z around 0 74.8%

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

6. Final simplification 74.8%

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

Developer Target 1: 96.5% accurate, 0.7× 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 2024137 
(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))))