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

Percentage Accurate: 82.5% → 96.0%

Time: 13.2s

Alternatives: 17

Speedup: 1.0×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z + 1}\\ t_1 := \frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+102}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{x\_m \cdot y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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))) (t_1 (/ (* x_m y_m) (* (+ z 1.0) (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 1e+102)
       (* t_0 (/ y_m (* z z)))
       (if (<= t_1 INFINITY)
         (/ (/ (* x_m y_m) z) z)
         (* (/ 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 t_1 = (x_m * y_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_1 <= 1e+102) {
		tmp = t_0 * (y_m / (z * z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_m * y_m) / z) / z;
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	return y_s * (x_s * tmp);
}

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 t_1 = (x_m * y_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_1 <= 1e+102) {
		tmp = t_0 * (y_m / (z * z));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_m * y_m) / z) / z;
	} 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)
	t_1 = (x_m * y_m) / ((z + 1.0) * (z * z))
	tmp = 0
	if t_1 <= 1e+102:
		tmp = t_0 * (y_m / (z * z))
	elif t_1 <= math.inf:
		tmp = ((x_m * y_m) / z) / z
	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))
	t_1 = Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z)))
	tmp = 0.0
	if (t_1 <= 1e+102)
		tmp = Float64(t_0 * Float64(y_m / Float64(z * z)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_m * y_m) / z) / z);
	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);
	t_1 = (x_m * y_m) / ((z + 1.0) * (z * z));
	tmp = 0.0;
	if (t_1 <= 1e+102)
		tmp = t_0 * (y_m / (z * z));
	elseif (t_1 <= Inf)
		tmp = ((x_m * y_m) / z) / z;
	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]}, Block[{t$95$1 = N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 1e+102], N[(t$95$0 * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 9.99999999999999977e101

   1. Initial program 92.5%

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

      1. sqr-neg 92.5%

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

      2. *-commutative 92.5%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

   if 9.99999999999999977e101 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < +inf.0

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 86.2%

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

      3. *-commutative 86.2%

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

      4. sqr-neg 86.2%

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

      5. *-commutative 86.2%

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

      6. distribute-rgt1-in 86.2%

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

      7. sqr-neg 86.2%

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

      8. fma-def 86.2%

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

      9. sqr-neg 86.2%

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

      10. cube-unmult 86.2%

          \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-def 86.2%

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

      2. associate-*r/ 88.2%

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

      3. *-commutative 88.2%

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

      4. cube-mult 88.2%

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

      5. distribute-rgt1-in 88.2%

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

      6. *-commutative 88.2%

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

      7. frac-times 86.2%

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

      8. associate-*l/ 88.2%

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

      9. associate-/r* 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in z around 0 98.2%

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

   if +inf.0 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. frac-times 33.4%

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

      3. associate-*l/ 23.2%

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

      4. times-frac 97.3%

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

   4. Applied egg-rr 97.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{+102}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 85.1%

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

      1. *-commutative 85.1%

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

      2. frac-times 92.1%

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

      3. associate-*l/ 91.9%

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

      4. times-frac 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in z around inf 95.3%

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

   if -1 < z < 1

   1. Initial program 81.4%

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

      1. sqr-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 81.9%

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

      4. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. associate-/r* 91.3%

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

      2. associate-*l/ 95.8%

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

      3. remove-double-neg 95.8%

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

      4. distribute-rgt-neg-out 95.8%

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

      5. add-sqr-sqrt 51.1%

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

      6. sqrt-prod 43.8%

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

      7. sqr-neg 43.8%

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

      8. sqrt-unprod 1.0%

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

      9. add-sqr-sqrt 1.4%

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

      10. frac-2neg 1.4%

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

      11. associate-*l/ 1.6%

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

      12. associate-/l* 1.4%

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

      13. add-sqr-sqrt 1.1%

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

      14. sqrt-unprod 35.8%

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

      15. sqr-neg 35.8%

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

      16. sqrt-unprod 46.7%

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

      17. add-sqr-sqrt 95.9%

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

   7. Applied egg-rr 95.9%

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

4. Final simplification 95.6%

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

Alternative 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.85\right):\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 0.85)))
     (/ (* (/ 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 <= 0.85)) {
		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 <= 0.85d0))) 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 <= 0.85)) {
		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 <= 0.85):
		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 <= 0.85))
		tmp = Float64(Float64(Float64(y_m / z) * 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 <= 0.85)))
		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, 0.85]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$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 0.849999999999999978 < z

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. sqr-neg 87.0%

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

      5. *-commutative 87.0%

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

      6. distribute-rgt1-in 66.5%

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

      7. sqr-neg 66.5%

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

      8. fma-def 87.0%

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

      9. sqr-neg 87.0%

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

      10. cube-unmult 87.0%

          \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-def 66.5%

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

      2. associate-*r/ 66.8%

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

      3. *-commutative 66.8%

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

      4. cube-mult 66.9%

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

      5. distribute-rgt1-in 85.1%

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

      6. *-commutative 85.1%

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

      7. frac-times 92.1%

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

      8. associate-/r* 96.3%

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

      9. associate-*l/ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in z around inf 96.4%

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

   if -1 < z < 0.849999999999999978

   1. Initial program 81.4%

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

      1. sqr-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 81.9%

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

      4. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. associate-/r* 91.3%

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

      2. associate-*l/ 95.8%

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

      3. remove-double-neg 95.8%

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

      4. distribute-rgt-neg-out 95.8%

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

      5. add-sqr-sqrt 51.1%

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

      6. sqrt-prod 43.8%

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

      7. sqr-neg 43.8%

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

      8. sqrt-unprod 1.0%

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

      9. add-sqr-sqrt 1.4%

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

      10. frac-2neg 1.4%

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

      11. associate-*l/ 1.6%

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

      12. associate-/l* 1.4%

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

      13. add-sqr-sqrt 1.1%

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

      14. sqrt-unprod 35.8%

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

      15. sqr-neg 35.8%

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

      16. sqrt-unprod 46.7%

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

      17. add-sqr-sqrt 95.9%

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

   7. Applied egg-rr 95.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.85\right):\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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) (/ z y_m)) 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) / (z / y_m)) / 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) / (z / y_m)) / 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) / (z / y_m)) / 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) / (z / y_m)) / 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(z / y_m)) / 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) / (z / y_m)) / 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[(z / y$95$m), $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 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. sqr-neg 87.0%

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

      5. *-commutative 87.0%

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

      6. distribute-rgt1-in 66.5%

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

      7. sqr-neg 66.5%

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

      8. fma-def 87.0%

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

      9. sqr-neg 87.0%

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

      10. cube-unmult 87.0%

          \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-def 66.5%

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

      2. associate-*r/ 66.8%

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

      3. *-commutative 66.8%

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

      4. cube-mult 66.9%

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

      5. distribute-rgt1-in 85.1%

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

      6. *-commutative 85.1%

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

      7. frac-times 92.1%

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

      8. associate-/r* 96.3%

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

      9. associate-*l/ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in z around inf 96.4%

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

      1. *-commutative 54.7%

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

      2. clear-num 54.7%

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

      3. un-div-inv 54.7%

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

   9. Applied egg-rr 96.5%

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

   if -1 < z < 1

   1. Initial program 81.4%

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

      1. sqr-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 81.9%

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

      4. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. associate-/r* 91.3%

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

      2. associate-*l/ 95.8%

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

      3. remove-double-neg 95.8%

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

      4. distribute-rgt-neg-out 95.8%

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

      5. add-sqr-sqrt 51.1%

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

      6. sqrt-prod 43.8%

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

      7. sqr-neg 43.8%

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

      8. sqrt-unprod 1.0%

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

      9. add-sqr-sqrt 1.4%

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

      10. frac-2neg 1.4%

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

      11. associate-*l/ 1.6%

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

      12. associate-/l* 1.4%

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

      13. add-sqr-sqrt 1.1%

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

      14. sqrt-unprod 35.8%

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

      15. sqr-neg 35.8%

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

      16. sqrt-unprod 46.7%

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

      17. add-sqr-sqrt 95.9%

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

   7. Applied egg-rr 95.9%

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

4. Final simplification 96.2%

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

Alternative 5: 94.5% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (* (/ y_m (* z z)) (/ x_m z))
     (if (<= z 1.0) (/ (/ y_m (/ z x_m)) z) (* (/ y_m z) (/ (/ 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 (z <= -1.0) {
		tmp = (y_m / (z * z)) * (x_m / z);
	} else if (z <= 1.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = (y_m / z) * ((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 (z <= (-1.0d0)) then
        tmp = (y_m / (z * z)) * (x_m / z)
    else if (z <= 1.0d0) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = (y_m / z) * ((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 (z <= -1.0) {
		tmp = (y_m / (z * z)) * (x_m / z);
	} else if (z <= 1.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = (y_m / z) * ((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 z <= -1.0:
		tmp = (y_m / (z * z)) * (x_m / z)
	elif z <= 1.0:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = (y_m / z) * ((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 (z <= -1.0)
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / z));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / 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 (z <= -1.0)
		tmp = (y_m / (z * z)) * (x_m / z);
	elseif (z <= 1.0)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = (y_m / z) * ((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[z, -1.0], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. times-frac 94.8%

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

      4. sqr-neg 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in z around inf 91.7%

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

   if -1 < z < 1

   1. Initial program 81.4%

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

      1. sqr-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 81.9%

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

      4. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. associate-/r* 91.3%

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

      2. associate-*l/ 95.8%

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

      3. remove-double-neg 95.8%

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

      4. distribute-rgt-neg-out 95.8%

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

      5. add-sqr-sqrt 51.1%

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

      6. sqrt-prod 43.8%

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

      7. sqr-neg 43.8%

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

      8. sqrt-unprod 1.0%

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

      9. add-sqr-sqrt 1.4%

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

      10. frac-2neg 1.4%

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

      11. associate-*l/ 1.6%

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

      12. associate-/l* 1.4%

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

      13. add-sqr-sqrt 1.1%

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

      14. sqrt-unprod 35.8%

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

      15. sqr-neg 35.8%

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

      16. sqrt-unprod 46.7%

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

      17. add-sqr-sqrt 95.9%

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

   7. Applied egg-rr 95.9%

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

   if 1 < z

   1. Initial program 84.7%

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

      1. *-commutative 84.7%

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

      2. frac-times 90.2%

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

      3. associate-*l/ 90.1%

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

      4. times-frac 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in z around inf 96.8%

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

4. Final simplification 95.2%

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

Alternative 6: 96.2% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (/ (/ (/ x_m z) (/ z y_m)) z)
     (if (<= z 1.0) (/ (/ y_m (/ z x_m)) z) (/ (/ (/ y_m z) (/ 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) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else if (z <= 1.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = ((y_m / z) / (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)) then
        tmp = ((x_m / z) / (z / y_m)) / z
    else if (z <= 1.0d0) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = ((y_m / z) / (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) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else if (z <= 1.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = ((y_m / z) / (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:
		tmp = ((x_m / z) / (z / y_m)) / z
	elif z <= 1.0:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = ((y_m / z) / (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)
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / y_m)) / z);
	elseif (z <= 1.0)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(y_m / z) / 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)
		tmp = ((x_m / z) / (z / y_m)) / z;
	elseif (z <= 1.0)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = ((y_m / z) / (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[LessEqual[z, -1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l/ 89.5%

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

      3. *-commutative 89.5%

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

      4. sqr-neg 89.5%

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

      5. *-commutative 89.5%

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

      6. distribute-rgt1-in 40.3%

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

      7. sqr-neg 40.3%

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

      8. fma-def 89.5%

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

      9. sqr-neg 89.5%

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

      10. cube-unmult 89.6%

          \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-def 40.3%

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

      2. associate-*r/ 41.8%

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

      3. *-commutative 41.8%

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

      4. cube-mult 41.9%

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

      5. distribute-rgt1-in 85.6%

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

      6. *-commutative 85.6%

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

      7. frac-times 94.8%

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

      8. associate-/r* 96.5%

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

      9. associate-*l/ 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in z around inf 94.9%

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

      1. *-commutative 52.4%

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

      2. clear-num 52.4%

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

      3. un-div-inv 52.4%

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

   9. Applied egg-rr 95.0%

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

   if -1 < z < 1

   1. Initial program 81.4%

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

      1. sqr-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 81.9%

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

      4. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around 0 80.2%

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

      1. associate-/r* 91.3%

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

      2. associate-*l/ 95.8%

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

      3. remove-double-neg 95.8%

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

      4. distribute-rgt-neg-out 95.8%

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

      5. add-sqr-sqrt 51.1%

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

      6. sqrt-prod 43.8%

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

      7. sqr-neg 43.8%

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

      8. sqrt-unprod 1.0%

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

      9. add-sqr-sqrt 1.4%

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

      10. frac-2neg 1.4%

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

      11. associate-*l/ 1.6%

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

      12. associate-/l* 1.4%

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

      13. add-sqr-sqrt 1.1%

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

      14. sqrt-unprod 35.8%

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

      15. sqr-neg 35.8%

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

      16. sqrt-unprod 46.7%

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

      17. add-sqr-sqrt 95.9%

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

   7. Applied egg-rr 95.9%

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

   if 1 < z

   1. Initial program 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-*l/ 85.2%

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

      3. *-commutative 85.2%

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

      4. sqr-neg 85.2%

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

      5. *-commutative 85.2%

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

      6. distribute-rgt1-in 85.2%

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

      7. sqr-neg 85.2%

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

      8. fma-def 85.2%

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

      9. sqr-neg 85.2%

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

      10. cube-unmult 85.2%

          \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-def 85.2%

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

      2. associate-*r/ 84.6%

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

      3. *-commutative 84.6%

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

      4. cube-mult 84.7%

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

      5. distribute-rgt1-in 84.7%

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

      6. *-commutative 84.7%

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

      7. frac-times 90.2%

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

      8. associate-/r* 96.1%

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

      9. associate-*l/ 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. clear-num 98.6%

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

      2. un-div-inv 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in z around inf 97.5%

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

4. Final simplification 96.2%

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

Alternative 7: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.6 \cdot 10^{+72}\right):\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 3.6e+72)))
     (* y_m (/ (/ 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 <= 3.6e+72)) {
		tmp = y_m * ((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 <= 3.6d+72))) then
        tmp = y_m * ((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 <= 3.6e+72)) {
		tmp = y_m * ((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 <= 3.6e+72):
		tmp = y_m * ((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 <= 3.6e+72))
		tmp = Float64(y_m * Float64(Float64(x_m / z) / Float64(-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 <= 3.6e+72)))
		tmp = y_m * ((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, 3.6e+72]], $MachinePrecision]], N[(y$95$m * 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 3.60000000000000035e72 < z

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. frac-times 91.7%

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

      3. associate-*l/ 91.5%

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

      4. times-frac 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in z around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. clear-num 57.2%

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

      3. un-div-inv 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. frac-2neg 57.2%

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

      2. associate-/r/ 62.2%

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

      3. add-sqr-sqrt 30.8%

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

      4. sqrt-unprod 54.2%

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

      5. sqr-neg 54.2%

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

      6. sqrt-unprod 31.7%

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

      7. add-sqr-sqrt 63.0%

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

   9. Applied egg-rr 63.0%

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

   if -1 < z < 3.60000000000000035e72

   1. Initial program 83.1%

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

      1. sqr-neg 83.1%

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

      2. *-commutative 83.1%

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

      3. times-frac 83.5%

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

      4. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in z around 0 75.0%

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

      1. associate-/r* 84.7%

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

      2. associate-*l/ 88.5%

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

      3. remove-double-neg 88.5%

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

      4. distribute-rgt-neg-out 88.5%

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

      5. add-sqr-sqrt 49.5%

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

      6. sqrt-prod 43.2%

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

      7. sqr-neg 43.2%

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

      8. sqrt-unprod 0.9%

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

      9. add-sqr-sqrt 4.9%

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

      10. frac-2neg 4.9%

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

      11. associate-*l/ 5.0%

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

      12. associate-/l* 4.9%

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

      13. add-sqr-sqrt 2.4%

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

      14. sqrt-unprod 35.3%

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

      15. sqr-neg 35.3%

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

      16. sqrt-unprod 43.4%

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

      17. add-sqr-sqrt 88.6%

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.6 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -5 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 -5e-310)))
     (/ y_m (/ z x_m))
     (* x_m (/ (- 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) {
	double tmp;
	if ((z <= -1.0) || !(z <= -5e-310)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m * (-y_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 <= (-5d-310)))) then
        tmp = y_m / (z / x_m)
    else
        tmp = x_m * (-y_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 <= -5e-310)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m * (-y_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 <= -5e-310):
		tmp = y_m / (z / x_m)
	else:
		tmp = x_m * (-y_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 <= -5e-310))
		tmp = Float64(y_m / Float64(z / x_m));
	else
		tmp = Float64(x_m * Float64(Float64(-y_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 <= -5e-310)))
		tmp = y_m / (z / x_m);
	else
		tmp = x_m * (-y_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, -5e-310]], $MachinePrecision]], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[((-y$95$m) / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or -4.999999999999985e-310 < z

   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 88.6%

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

      3. associate-*l/ 87.8%

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

      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 0 57.9%

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

      1. +-commutative 57.9%

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

      2. mul-1-neg 57.9%

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

      3. unsub-neg 57.9%

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

   7. Simplified 57.9%

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

   8. Taylor expanded in z around inf 21.7%

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

      1. mul-1-neg 21.7%

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

      2. associate-*l/ 23.7%

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

      3. distribute-rgt-neg-out 23.7%

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

   10. Simplified 23.7%

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

       1. *-commutative 23.7%

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

       2. clear-num 25.2%

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

       3. un-div-inv 25.2%

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

       4. add-sqr-sqrt 12.1%

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

       5. sqrt-unprod 29.7%

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

       6. sqr-neg 29.7%

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

       7. sqrt-unprod 18.5%

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

       8. add-sqr-sqrt 41.0%

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

   12. Applied egg-rr 41.0%

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

   if -1 < z < -4.999999999999985e-310

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. frac-times 82.3%

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

      3. associate-*l/ 83.5%

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

      4. times-frac 93.2%

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

   4. Applied egg-rr 93.2%

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

   5. Taylor expanded in z around 0 93.0%

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

      1. +-commutative 93.0%

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

      2. mul-1-neg 93.0%

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

      3. unsub-neg 93.0%

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

   7. Simplified 93.0%

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

   8. Taylor expanded in z around inf 35.5%

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

      1. neg-mul-1 35.5%

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

   10. Simplified 35.5%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -5 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 1.75e-131) (* (/ 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 <= 1.75e-131) {
		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 <= 1.75d-131) 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 <= 1.75e-131) {
		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 <= 1.75e-131:
		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 (x_m <= 1.75e-131)
		tmp = Float64(Float64(y_m / z) * Float64(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 <= 1.75e-131)
		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[x$95$m, 1.75e-131], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$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 < 1.7500000000000001e-131

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. frac-times 86.5%

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

      3. associate-*l/ 86.0%

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

      4. times-frac 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in z around 0 77.3%

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

   if 1.7500000000000001e-131 < x

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. *-commutative 83.8%

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

      3. times-frac 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in z around 0 69.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 9.5e-133) (/ (/ x_m z) (/ z y_m)) (* 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 <= 9.5e-133) {
		tmp = (x_m / z) / (z / y_m);
	} 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 <= 9.5d-133) then
        tmp = (x_m / z) / (z / y_m)
    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 <= 9.5e-133) {
		tmp = (x_m / z) / (z / y_m);
	} 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 <= 9.5e-133:
		tmp = (x_m / z) / (z / y_m)
	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 <= 9.5e-133)
		tmp = Float64(Float64(x_m / z) / Float64(z / y_m));
	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 <= 9.5e-133)
		tmp = (x_m / z) / (z / y_m);
	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, 9.5e-133], N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $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 < 9.4999999999999992e-133

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. frac-times 86.5%

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

      3. associate-*l/ 86.0%

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

      4. times-frac 95.8%

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

   4. Applied egg-rr 95.8%

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

   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 77.2%

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

      3. un-div-inv 77.7%

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

   7. Applied egg-rr 77.7%

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

   if 9.4999999999999992e-133 < x

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. *-commutative 83.8%

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

      3. times-frac 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in z around 0 69.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 2.2e-131) (/ (/ 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 <= 2.2e-131) {
		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 <= 2.2d-131) 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 <= 2.2e-131) {
		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 <= 2.2e-131:
		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 (x_m <= 2.2e-131)
		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 <= 2.2e-131)
		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[x$95$m, 2.2e-131], 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 x < 2.2e-131

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. *-commutative 83.0%

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

      3. times-frac 86.5%

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

      4. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around 0 72.7%

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

      1. associate-/r* 76.1%

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

      2. associate-*l/ 77.1%

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

      3. remove-double-neg 77.1%

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

      4. distribute-rgt-neg-out 77.1%

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

      5. add-sqr-sqrt 40.9%

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

      6. sqrt-prod 53.1%

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

      7. sqr-neg 53.1%

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

      8. sqrt-unprod 13.5%

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

      9. add-sqr-sqrt 31.0%

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

      10. frac-2neg 31.0%

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

      11. associate-*l/ 30.5%

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

      12. associate-/l* 31.6%

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

      13. add-sqr-sqrt 23.9%

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

      14. sqrt-unprod 29.8%

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

      15. sqr-neg 29.8%

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

      16. sqrt-unprod 17.1%

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

      17. add-sqr-sqrt 78.6%

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

   7. Applied egg-rr 78.6%

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

   if 2.2e-131 < x

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. *-commutative 83.8%

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

      3. times-frac 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in z around 0 69.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z -5e-310) (* (/ x_m z) (- y_m)) (/ y_m (/ z x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = (x_m / z) * -y_m;
	} else {
		tmp = y_m / (z / x_m);
	}
	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 <= (-5d-310)) then
        tmp = (x_m / z) * -y_m
    else
        tmp = y_m / (z / x_m)
    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 <= -5e-310) {
		tmp = (x_m / z) * -y_m;
	} else {
		tmp = y_m / (z / x_m);
	}
	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 <= -5e-310:
		tmp = (x_m / z) * -y_m
	else:
		tmp = y_m / (z / x_m)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(Float64(x_m / z) * Float64(-y_m));
	else
		tmp = Float64(y_m / Float64(z / x_m));
	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 <= -5e-310)
		tmp = (x_m / z) * -y_m;
	else
		tmp = y_m / (z / x_m);
	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[z, -5e-310], N[(N[(x$95$m / z), $MachinePrecision] * (-y$95$m)), $MachinePrecision], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. frac-times 88.5%

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

      3. associate-*l/ 88.9%

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

      4. times-frac 94.7%

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

   4. Applied egg-rr 94.7%

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

   5. Taylor expanded in z around 0 65.7%

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

      1. +-commutative 65.7%

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

      2. mul-1-neg 65.7%

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

      3. unsub-neg 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in z around inf 30.5%

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

      1. mul-1-neg 30.5%

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

      2. associate-*l/ 35.0%

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

      3. distribute-rgt-neg-out 35.0%

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

   10. Simplified 35.0%

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

   if -4.999999999999985e-310 < z

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-times 86.2%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in z around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. mul-1-neg 65.8%

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

      3. unsub-neg 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in z around inf 17.7%

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

      1. mul-1-neg 17.7%

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

      2. associate-*l/ 17.7%

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

      3. distribute-rgt-neg-out 17.7%

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

   10. Simplified 17.7%

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

       1. *-commutative 17.7%

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

       2. clear-num 18.3%

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

       3. un-div-inv 18.3%

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

       4. add-sqr-sqrt 7.2%

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

       5. sqrt-unprod 26.6%

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

       6. sqr-neg 26.6%

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

       7. sqrt-unprod 18.3%

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

       8. add-sqr-sqrt 39.6%

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

   12. Applied egg-rr 39.6%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.2% 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 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 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 87.2%

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

   3. associate-*l/ 86.9%

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

   4. times-frac 95.9%

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

4. Applied egg-rr 95.9%

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

5. Final simplification 95.9%

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

Alternative 14: 96.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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(x_m / Float64(Float64(z / 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[(x$95$m / N[(N[(z / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. 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. associate-*l/ 84.5%

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

   3. *-commutative 84.5%

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

   4. sqr-neg 84.5%

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

   5. *-commutative 84.5%

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

   6. distribute-rgt1-in 74.0%

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

   7. sqr-neg 74.0%

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

   8. fma-def 84.5%

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

   9. sqr-neg 84.5%

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

   10. cube-unmult 84.5%

       \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-def 73.9%

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

   2. associate-*r/ 73.9%

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

   3. *-commutative 73.9%

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

   4. cube-mult 73.9%

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

   5. distribute-rgt1-in 83.3%

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

   6. *-commutative 83.3%

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

   7. frac-times 87.2%

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

   8. associate-/r* 94.7%

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

   9. associate-*l/ 97.8%

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

6. Applied egg-rr 97.8%

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

   1. clear-num 97.8%

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

   2. frac-times 96.3%

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

   3. *-un-lft-identity 96.3%

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

8. Applied egg-rr 96.3%

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

9. Final simplification 96.3%

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

Alternative 15: 95.8% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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(Float64(y_m / z) * 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[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. 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. associate-*l/ 84.5%

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

   3. *-commutative 84.5%

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

   4. sqr-neg 84.5%

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

   5. *-commutative 84.5%

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

   6. distribute-rgt1-in 74.0%

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

   7. sqr-neg 74.0%

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

   8. fma-def 84.5%

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

   9. sqr-neg 84.5%

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

   10. cube-unmult 84.5%

       \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-def 73.9%

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

   2. associate-*r/ 73.9%

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

   3. *-commutative 73.9%

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

   4. cube-mult 73.9%

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

   5. distribute-rgt1-in 83.3%

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

   6. *-commutative 83.3%

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

   7. frac-times 87.2%

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

   8. associate-/r* 94.7%

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

   9. associate-*l/ 97.8%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 16: 73.8% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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)))))

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)));
}

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)))
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)));
}

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)))

Julia

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

Wolfram

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

Derivation

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 87.2%

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

   3. associate-*l/ 86.9%

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

   4. times-frac 95.9%

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

4. Applied egg-rr 95.9%

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

5. Taylor expanded in z around 0 73.2%

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

6. Final simplification 73.2%

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

Alternative 17: 33.4% accurate, 2.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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)))))

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)));
}

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)))
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)));
}

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)))

Julia

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

Wolfram

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

Derivation

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 87.2%

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

   3. associate-*l/ 86.9%

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

   4. times-frac 95.9%

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

4. Applied egg-rr 95.9%

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

5. Taylor expanded in z around 0 65.7%

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

   1. +-commutative 65.7%

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

   2. mul-1-neg 65.7%

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

   3. unsub-neg 65.7%

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

7. Simplified 65.7%

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

8. Taylor expanded in z around inf 23.3%

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

   1. mul-1-neg 23.3%

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

   2. associate-*l/ 25.3%

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

   3. distribute-rgt-neg-out 25.3%

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

10. Simplified 25.3%

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

    1. *-commutative 25.3%

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

    2. clear-num 26.4%

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

    3. un-div-inv 26.4%

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

    4. add-sqr-sqrt 13.0%

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

    5. sqrt-unprod 27.0%

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

    6. sqr-neg 27.0%

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

    7. sqrt-unprod 14.8%

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

    8. add-sqr-sqrt 32.5%

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

12. Applied egg-rr 32.5%

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

13. Final simplification 32.5%

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

Developer target: 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 2024027 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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