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

Percentage Accurate: 83.1% → 97.1%

Time: 10.8s

Alternatives: 15

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: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\frac{y\_m}{z + 1} \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) 4e+75)
     (* (/ y_m (+ z 1.0)) (/ (/ x_m z) z))
     (/ (/ (/ x_m z) (/ z y_m)) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 4e+75) {
		tmp = (y_m / (z + 1.0)) * ((x_m / z) / z);
	} else {
		tmp = ((x_m / z) / (z / 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) * (z * z)) <= 4d+75) then
        tmp = (y_m / (z + 1.0d0)) * ((x_m / z) / z)
    else
        tmp = ((x_m / z) / (z / 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 * z)) <= 4e+75) {
		tmp = (y_m / (z + 1.0)) * ((x_m / z) / z);
	} else {
		tmp = ((x_m / z) / (z / 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) * (z * z)) <= 4e+75:
		tmp = (y_m / (z + 1.0)) * ((x_m / z) / z)
	else:
		tmp = ((x_m / z) / (z / 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 (Float64(Float64(z + 1.0) * Float64(z * z)) <= 4e+75)
		tmp = Float64(Float64(y_m / Float64(z + 1.0)) * Float64(Float64(x_m / z) / z));
	else
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / 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 * z)) <= 4e+75)
		tmp = (y_m / (z + 1.0)) * ((x_m / z) / z);
	else
		tmp = ((x_m / z) / (z / 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[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 4e+75], N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 3.99999999999999971e75

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-/l* 83.0%

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

      3. sqr-neg 83.0%

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

      4. associate-/r* 84.4%

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

      5. sqr-neg 84.4%

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

   3. Simplified 84.4%

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

      1. associate-*r/ 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-*r/ 85.8%

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

      4. associate-/r* 92.3%

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

      5. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

      2. associate-/l* 92.3%

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

   8. Applied egg-rr 92.3%

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

   if 3.99999999999999971e75 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-/l* 81.0%

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

      3. sqr-neg 81.0%

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

      4. associate-/r* 86.4%

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

      5. sqr-neg 86.4%

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

   3. Simplified 86.4%

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

      1. associate-*r/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-*r/ 88.1%

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

      4. associate-/r* 91.2%

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

      5. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in z around inf 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\_m\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \left(-x\_m\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-142}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (* (/ x_m z) y_m)
     (if (<= z -5e-310)
       (* (/ y_m z) (- x_m))
       (if (<= z 3.2e-142) (* x_m (/ y_m z)) (/ 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 <= -1.0) {
		tmp = (x_m / z) * y_m;
	} else if (z <= -5e-310) {
		tmp = (y_m / z) * -x_m;
	} else if (z <= 3.2e-142) {
		tmp = x_m * (y_m / z);
	} 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 <= (-1.0d0)) then
        tmp = (x_m / z) * y_m
    else if (z <= (-5d-310)) then
        tmp = (y_m / z) * -x_m
    else if (z <= 3.2d-142) then
        tmp = x_m * (y_m / z)
    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 <= -1.0) {
		tmp = (x_m / z) * y_m;
	} else if (z <= -5e-310) {
		tmp = (y_m / z) * -x_m;
	} else if (z <= 3.2e-142) {
		tmp = x_m * (y_m / z);
	} 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 <= -1.0:
		tmp = (x_m / z) * y_m
	elif z <= -5e-310:
		tmp = (y_m / z) * -x_m
	elif z <= 3.2e-142:
		tmp = x_m * (y_m / z)
	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 <= -1.0)
		tmp = Float64(Float64(x_m / z) * y_m);
	elseif (z <= -5e-310)
		tmp = Float64(Float64(y_m / z) * Float64(-x_m));
	elseif (z <= 3.2e-142)
		tmp = Float64(x_m * Float64(y_m / z));
	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 <= -1.0)
		tmp = (x_m / z) * y_m;
	elseif (z <= -5e-310)
		tmp = (y_m / z) * -x_m;
	elseif (z <= 3.2e-142)
		tmp = x_m * (y_m / z);
	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, -1.0], N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[z, -5e-310], N[(N[(y$95$m / z), $MachinePrecision] * (-x$95$m)), $MachinePrecision], If[LessEqual[z, 3.2e-142], N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. sqr-neg 79.7%

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

      3. times-frac 96.1%

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

      4. sqr-neg 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in z around 0 48.3%

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

      1. mul-1-neg 48.3%

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

      2. unsub-neg 48.3%

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

   7. Simplified 48.3%

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

   8. Taylor expanded in z around inf 37.4%

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

      1. mul-1-neg 37.4%

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

      2. associate-*l/ 41.7%

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

      3. *-commutative 41.7%

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

      4. distribute-rgt-neg-in 41.7%

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

      5. distribute-frac-neg2 41.7%

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

   10. Simplified 41.7%

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

       1. add-sqr-sqrt 29.1%

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

       2. sqrt-unprod 56.1%

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

       3. distribute-frac-neg2 56.1%

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

       4. distribute-frac-neg2 56.1%

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

       5. sqr-neg 56.1%

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

       6. sqrt-unprod 35.4%

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

       7. add-sqr-sqrt 44.3%

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

       8. *-un-lft-identity 44.3%

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

   12. Applied egg-rr 44.3%

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

       1. *-lft-identity 44.3%

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

   14. Simplified 44.3%

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

   if -1 < z < -4.999999999999985e-310

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

      2. sqr-neg 73.5%

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

      3. times-frac 81.7%

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

      4. sqr-neg 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in z around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. unsub-neg 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in z around inf 44.5%

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

      1. mul-1-neg 44.5%

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

      2. associate-*r/ 47.8%

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

      3. distribute-lft-neg-out 47.8%

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

      4. *-commutative 47.8%

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

   10. Simplified 47.8%

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

   if -4.999999999999985e-310 < z < 3.1999999999999998e-142

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. sqr-neg 69.8%

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

      3. times-frac 71.3%

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

      4. sqr-neg 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in z around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. unsub-neg 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in z around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. associate-*l/ 0.5%

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

      3. *-commutative 0.5%

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

      4. distribute-rgt-neg-in 0.5%

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

      5. distribute-frac-neg2 0.5%

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

   10. Simplified 0.5%

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

       1. distribute-frac-neg2 0.5%

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

       2. distribute-frac-neg 0.5%

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

       3. associate-*r/ 0.6%

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

   12. Applied egg-rr 0.6%

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

       1. add-sqr-sqrt 0.6%

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

       2. times-frac 0.5%

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

       3. add-sqr-sqrt 0.4%

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

       4. sqrt-unprod 22.0%

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

       5. sqr-neg 22.0%

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

       6. sqrt-unprod 19.6%

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

       7. add-sqr-sqrt 32.9%

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

       8. times-frac 22.3%

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

       9. add-sqr-sqrt 22.3%

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

       10. clear-num 22.3%

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

       11. associate-/r* 35.2%

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

       12. associate-/r/ 37.3%

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

       13. clear-num 37.3%

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

   14. Applied egg-rr 37.3%

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

   if 3.1999999999999998e-142 < z

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. sqr-neg 85.5%

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

      3. times-frac 93.4%

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

      4. sqr-neg 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in z around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. unsub-neg 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in z around inf 19.0%

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

      1. mul-1-neg 19.0%

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

      2. associate-*l/ 20.2%

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

      3. *-commutative 20.2%

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

      4. distribute-rgt-neg-in 20.2%

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

      5. distribute-frac-neg2 20.2%

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

   10. Simplified 20.2%

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

       1. add-sqr-sqrt 18.0%

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

       2. sqrt-unprod 37.2%

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

       3. distribute-frac-neg2 37.2%

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

       4. distribute-frac-neg2 37.2%

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

       5. sqr-neg 37.2%

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

       6. sqrt-unprod 24.2%

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

       7. add-sqr-sqrt 33.7%

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

       8. clear-num 33.7%

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

       9. un-div-inv 33.7%

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

   12. Applied egg-rr 33.7%

       \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (/ (* (/ x_m z) (/ y_m z)) z)
     (/ y_m (* z (/ 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 <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = y_m / (z * (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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x_m / z) * (y_m / z)) / z
    else
        tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = y_m / (z * (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 <= -1.0) or not (z <= 1.0):
		tmp = ((x_m / z) * (y_m / z)) / z
	else:
		tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(y_m / Float64(z * 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 <= -1.0) || ~((z <= 1.0)))
		tmp = ((x_m / z) * (y_m / z)) / z;
	else
		tmp = y_m / (z * (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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/l* 84.8%

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

      3. sqr-neg 84.8%

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

      4. associate-/r* 89.4%

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

      5. sqr-neg 89.4%

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

   3. Simplified 89.4%

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

      1. associate-*r/ 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r/ 92.3%

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

      4. associate-/r* 95.2%

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

      5. associate-*l/ 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around inf 98.0%

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

   if -1 < z < 1

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/l* 80.6%

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

      3. sqr-neg 80.6%

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

      4. associate-/r* 80.6%

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

      5. sqr-neg 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*r/ 80.6%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. clear-num 89.1%

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

      2. un-div-inv 90.9%

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

      3. div-inv 90.9%

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

      4. clear-num 90.9%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in z around 0 89.7%

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

4. Final simplification 93.7%

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

Alternative 4: 94.4% accurate, 0.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ x_m z) (/ y_m (* z z)))
     (/ y_m (* z (/ 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 <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = y_m / (z * (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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x_m / z) * (y_m / (z * z))
    else
        tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = y_m / (z * (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 <= -1.0) or not (z <= 1.0):
		tmp = (x_m / z) * (y_m / (z * z))
	else:
		tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(y_m / Float64(z * 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 <= -1.0) || ~((z <= 1.0)))
		tmp = (x_m / z) * (y_m / (z * z));
	else
		tmp = y_m / (z * (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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. sqr-neg 80.7%

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

      3. times-frac 95.0%

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

      4. sqr-neg 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in z around inf 94.0%

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

   if -1 < z < 1

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/l* 80.6%

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

      3. sqr-neg 80.6%

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

      4. associate-/r* 80.6%

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

      5. sqr-neg 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*r/ 80.6%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. clear-num 89.1%

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

      2. un-div-inv 90.9%

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

      3. div-inv 90.9%

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

      4. clear-num 90.9%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in z around 0 89.7%

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

4. Final simplification 91.8%

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

Alternative 5: 90.6% accurate, 0.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (* y_m (/ (/ x_m (* z z)) z))
     (/ y_m (* z (/ 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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * ((x_m / (z * z)) / z);
	} else {
		tmp = y_m / (z * (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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = y_m * ((x_m / (z * z)) / z)
    else
        tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0)) {
		tmp = y_m * ((x_m / (z * z)) / z);
	} else {
		tmp = y_m / (z * (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 <= -1.0) or not (z <= 1.0):
		tmp = y_m * ((x_m / (z * z)) / z)
	else:
		tmp = y_m / (z * (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 <= -1.0) || !(z <= 1.0))
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / z));
	else
		tmp = Float64(y_m / Float64(z * 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 <= -1.0) || ~((z <= 1.0)))
		tmp = y_m * ((x_m / (z * z)) / z);
	else
		tmp = y_m / (z * (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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/l* 84.8%

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

      3. sqr-neg 84.8%

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

      4. associate-/r* 89.4%

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

      5. sqr-neg 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in z around inf 88.3%

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

   if -1 < z < 1

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/l* 80.6%

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

      3. sqr-neg 80.6%

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

      4. associate-/r* 80.6%

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

      5. sqr-neg 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*r/ 80.6%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. clear-num 89.1%

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

      2. un-div-inv 90.9%

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

      3. div-inv 90.9%

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

      4. clear-num 90.9%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in z around 0 89.7%

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

4. Final simplification 89.0%

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

Alternative 6: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x\_m}{z + 1}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -3.5e-7)
     (* (/ y_m (* z z)) (/ x_m (+ z 1.0)))
     (if (<= z 1.0) (/ y_m (* z (/ z x_m))) (/ (/ (/ x_m z) (/ z y_m)) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -3.5e-7) {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = ((x_m / z) / (z / 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 <= (-3.5d-7)) then
        tmp = (y_m / (z * z)) * (x_m / (z + 1.0d0))
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = ((x_m / z) / (z / 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 <= -3.5e-7) {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = ((x_m / z) / (z / 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 <= -3.5e-7:
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0))
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = ((x_m / z) / (z / 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 <= -3.5e-7)
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / Float64(z + 1.0)));
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / 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 <= -3.5e-7)
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = ((x_m / z) / (z / 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[LessEqual[z, -3.5e-7], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999984e-7

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. sqr-neg 80.0%

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

      3. times-frac 96.2%

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

      4. sqr-neg 96.2%

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

   3. Simplified 96.2%

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

   if -3.49999999999999984e-7 < z < 1

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/l* 80.5%

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

      3. sqr-neg 80.5%

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

      4. associate-/r* 80.5%

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

      5. sqr-neg 80.5%

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

   3. Simplified 80.5%

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

      1. associate-*r/ 80.5%

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

      2. *-commutative 80.5%

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

      3. associate-*r/ 80.5%

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

      4. associate-/r* 89.1%

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

      5. associate-*l/ 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/l* 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. clear-num 89.1%

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

      2. un-div-inv 90.9%

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

      3. div-inv 90.8%

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

      4. clear-num 90.9%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in z around 0 90.1%

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

   if 1 < z

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 82.0%

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

      3. sqr-neg 82.0%

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

      4. associate-/r* 87.2%

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

      5. sqr-neg 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*r/ 88.7%

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

      4. associate-/r* 91.7%

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

      5. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in z around inf 98.6%

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

Alternative 7: 96.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = ((x_m / z) / (z / 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)) then
        tmp = ((x_m / z) * (y_m / z)) / z
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = ((x_m / z) / (z / 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) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = ((x_m / z) / (z / 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:
		tmp = ((x_m / z) * (y_m / z)) / z
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = ((x_m / z) / (z / 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)
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / 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)
		tmp = ((x_m / z) * (y_m / z)) / z;
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = ((x_m / z) / (z / 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[LessEqual[z, -1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-/l* 87.2%

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

      3. sqr-neg 87.2%

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

      4. associate-/r* 91.3%

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

      5. sqr-neg 91.3%

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

   3. Simplified 91.3%

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

      1. associate-*r/ 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*r/ 95.4%

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

      4. associate-/r* 98.2%

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

      5. associate-*l/ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in z around inf 97.5%

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

   if -1 < z < 1

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/l* 80.6%

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

      3. sqr-neg 80.6%

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

      4. associate-/r* 80.6%

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

      5. sqr-neg 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*r/ 80.6%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. clear-num 89.1%

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

      2. un-div-inv 90.9%

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

      3. div-inv 90.9%

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

      4. clear-num 90.9%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in z around 0 89.7%

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

   if 1 < z

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 82.0%

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

      3. sqr-neg 82.0%

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

      4. associate-/r* 87.2%

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

      5. sqr-neg 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*r/ 88.7%

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

      4. associate-/r* 91.7%

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

      5. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in z around inf 98.6%

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

Alternative 8: 42.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(-y\_m\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-142}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -5e-310)
     (* (/ x_m z) (- y_m))
     (if (<= z 3.2e-142) (* x_m (/ y_m z)) (/ 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 if (z <= 3.2e-142) {
		tmp = x_m * (y_m / z);
	} 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 if (z <= 3.2d-142) then
        tmp = x_m * (y_m / z)
    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 if (z <= 3.2e-142) {
		tmp = x_m * (y_m / z);
	} 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
	elif z <= 3.2e-142:
		tmp = x_m * (y_m / z)
	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));
	elseif (z <= 3.2e-142)
		tmp = Float64(x_m * Float64(y_m / z));
	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;
	elseif (z <= 3.2e-142)
		tmp = x_m * (y_m / z);
	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], If[LessEqual[z, 3.2e-142], N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. sqr-neg 76.7%

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

      3. times-frac 89.3%

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

      4. sqr-neg 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in z around 0 63.8%

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

      1. mul-1-neg 63.8%

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

      2. unsub-neg 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in z around inf 40.8%

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

      1. mul-1-neg 40.8%

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

      2. associate-*l/ 45.3%

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

      3. *-commutative 45.3%

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

      4. distribute-rgt-neg-in 45.3%

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

      5. distribute-frac-neg2 45.3%

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

   10. Simplified 45.3%

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

   if -4.999999999999985e-310 < z < 3.1999999999999998e-142

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. sqr-neg 69.8%

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

      3. times-frac 71.3%

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

      4. sqr-neg 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in z around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. unsub-neg 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in z around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. associate-*l/ 0.5%

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

      3. *-commutative 0.5%

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

      4. distribute-rgt-neg-in 0.5%

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

      5. distribute-frac-neg2 0.5%

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

   10. Simplified 0.5%

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

       1. distribute-frac-neg2 0.5%

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

       2. distribute-frac-neg 0.5%

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

       3. associate-*r/ 0.6%

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

   12. Applied egg-rr 0.6%

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

       1. add-sqr-sqrt 0.6%

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

       2. times-frac 0.5%

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

       3. add-sqr-sqrt 0.4%

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

       4. sqrt-unprod 22.0%

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

       5. sqr-neg 22.0%

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

       6. sqrt-unprod 19.6%

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

       7. add-sqr-sqrt 32.9%

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

       8. times-frac 22.3%

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

       9. add-sqr-sqrt 22.3%

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

       10. clear-num 22.3%

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

       11. associate-/r* 35.2%

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

       12. associate-/r/ 37.3%

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

       13. clear-num 37.3%

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

   14. Applied egg-rr 37.3%

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

   if 3.1999999999999998e-142 < z

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. sqr-neg 85.5%

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

      3. times-frac 93.4%

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

      4. sqr-neg 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in z around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. unsub-neg 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in z around inf 19.0%

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

      1. mul-1-neg 19.0%

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

      2. associate-*l/ 20.2%

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

      3. *-commutative 20.2%

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

      4. distribute-rgt-neg-in 20.2%

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

      5. distribute-frac-neg2 20.2%

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

   10. Simplified 20.2%

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

       1. add-sqr-sqrt 18.0%

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

       2. sqrt-unprod 37.2%

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

       3. distribute-frac-neg2 37.2%

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

       4. distribute-frac-neg2 37.2%

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

       5. sqr-neg 37.2%

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

       6. sqrt-unprod 24.2%

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

       7. add-sqr-sqrt 33.7%

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

       8. clear-num 33.7%

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

       9. un-div-inv 33.7%

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

   12. Applied egg-rr 33.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\_m\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-168}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -6.8e-308)
     (* (/ x_m z) y_m)
     (if (<= z 1.04e-168) (* x_m (/ y_m z)) (/ 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 <= -6.8e-308) {
		tmp = (x_m / z) * y_m;
	} else if (z <= 1.04e-168) {
		tmp = x_m * (y_m / z);
	} 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 <= (-6.8d-308)) then
        tmp = (x_m / z) * y_m
    else if (z <= 1.04d-168) then
        tmp = x_m * (y_m / z)
    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 <= -6.8e-308) {
		tmp = (x_m / z) * y_m;
	} else if (z <= 1.04e-168) {
		tmp = x_m * (y_m / z);
	} 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 <= -6.8e-308:
		tmp = (x_m / z) * y_m
	elif z <= 1.04e-168:
		tmp = x_m * (y_m / z)
	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 <= -6.8e-308)
		tmp = Float64(Float64(x_m / z) * y_m);
	elseif (z <= 1.04e-168)
		tmp = Float64(x_m * Float64(y_m / z));
	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 <= -6.8e-308)
		tmp = (x_m / z) * y_m;
	elseif (z <= 1.04e-168)
		tmp = x_m * (y_m / z);
	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, -6.8e-308], N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[z, 1.04e-168], N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -6.79999999999999998e-308

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. sqr-neg 77.3%

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

      3. times-frac 89.9%

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

      4. sqr-neg 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in z around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unsub-neg 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in z around inf 41.1%

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

      1. mul-1-neg 41.1%

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

      2. associate-*l/ 45.6%

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

      3. *-commutative 45.6%

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

      4. distribute-rgt-neg-in 45.6%

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

      5. distribute-frac-neg2 45.6%

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

   10. Simplified 45.6%

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

       1. add-sqr-sqrt 27.0%

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

       2. sqrt-unprod 43.9%

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

       3. distribute-frac-neg2 43.9%

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

       4. distribute-frac-neg2 43.9%

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

       5. sqr-neg 43.9%

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

       6. sqrt-unprod 20.7%

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

       7. add-sqr-sqrt 25.6%

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

       8. *-un-lft-identity 25.6%

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

   12. Applied egg-rr 25.6%

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

       1. *-lft-identity 25.6%

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

   14. Simplified 25.6%

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

   if -6.79999999999999998e-308 < z < 1.04000000000000006e-168

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. sqr-neg 70.5%

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

      3. times-frac 70.9%

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

      4. sqr-neg 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in z around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. unsub-neg 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in z around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. associate-*l/ 0.5%

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

      3. *-commutative 0.5%

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

      4. distribute-rgt-neg-in 0.5%

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

      5. distribute-frac-neg2 0.5%

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

   10. Simplified 0.5%

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

       1. distribute-frac-neg2 0.5%

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

       2. distribute-frac-neg 0.5%

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

       3. associate-*r/ 0.6%

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

   12. Applied egg-rr 0.6%

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

       1. add-sqr-sqrt 0.5%

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

       2. times-frac 0.5%

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

       3. add-sqr-sqrt 0.3%

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

       4. sqrt-unprod 22.9%

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

       5. sqr-neg 22.9%

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

       6. sqrt-unprod 20.5%

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

       7. add-sqr-sqrt 34.2%

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

       8. times-frac 23.1%

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

       9. add-sqr-sqrt 23.1%

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

       10. clear-num 23.1%

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

       11. associate-/r* 36.5%

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

       12. associate-/r/ 38.8%

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

       13. clear-num 38.8%

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

   14. Applied egg-rr 38.8%

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

   if 1.04000000000000006e-168 < z

   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. *-commutative 83.8%

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

      2. sqr-neg 83.8%

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

      3. times-frac 92.1%

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

      4. sqr-neg 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in z around 0 60.4%

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

      1. mul-1-neg 60.4%

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

      2. unsub-neg 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in z around inf 18.4%

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

      1. mul-1-neg 18.4%

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

      2. associate-*l/ 19.5%

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

      3. *-commutative 19.5%

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

      4. distribute-rgt-neg-in 19.5%

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

      5. distribute-frac-neg2 19.5%

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

   10. Simplified 19.5%

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

       1. add-sqr-sqrt 17.4%

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

       2. sqrt-unprod 36.0%

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

       3. distribute-frac-neg2 36.0%

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

       4. distribute-frac-neg2 36.0%

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

       5. sqr-neg 36.0%

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

       6. sqrt-unprod 23.4%

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

       7. add-sqr-sqrt 32.7%

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

       8. clear-num 32.7%

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

       9. un-div-inv 32.7%

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

   12. Applied egg-rr 32.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-/l* 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-/r* 84.8%

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

   5. sqr-neg 84.8%

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

3. Simplified 84.8%

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

   1. associate-*r/ 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*r/ 86.2%

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

   4. associate-/r* 92.1%

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

   5. associate-*l/ 97.1%

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

6. Applied egg-rr 97.1%

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

   1. clear-num 97.1%

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

   2. un-div-inv 97.1%

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

8. Applied egg-rr 97.1%

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

Alternative 11: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-/l* 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-/r* 84.8%

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

   5. sqr-neg 84.8%

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

3. Simplified 84.8%

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

   1. associate-*r/ 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*r/ 86.2%

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

   4. associate-/r* 92.1%

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

   5. associate-*l/ 97.1%

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

6. Applied egg-rr 97.1%

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

Alternative 12: 31.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z -6.8e-308) (* (/ x_m z) y_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 <= -6.8e-308) {
		tmp = (x_m / z) * y_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 <= (-6.8d-308)) then
        tmp = (x_m / z) * y_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 <= -6.8e-308) {
		tmp = (x_m / z) * y_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 <= -6.8e-308:
		tmp = (x_m / z) * y_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 <= -6.8e-308)
		tmp = Float64(Float64(x_m / z) * y_m);
	else
		tmp = Float64(x_m * 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 <= -6.8e-308)
		tmp = (x_m / z) * y_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[LessEqual[z, -6.8e-308], N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.79999999999999998e-308

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. sqr-neg 77.3%

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

      3. times-frac 89.9%

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

      4. sqr-neg 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in z around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unsub-neg 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in z around inf 41.1%

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

      1. mul-1-neg 41.1%

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

      2. associate-*l/ 45.6%

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

      3. *-commutative 45.6%

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

      4. distribute-rgt-neg-in 45.6%

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

      5. distribute-frac-neg2 45.6%

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

   10. Simplified 45.6%

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

       1. add-sqr-sqrt 27.0%

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

       2. sqrt-unprod 43.9%

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

       3. distribute-frac-neg2 43.9%

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

       4. distribute-frac-neg2 43.9%

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

       5. sqr-neg 43.9%

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

       6. sqrt-unprod 20.7%

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

       7. add-sqr-sqrt 25.6%

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

       8. *-un-lft-identity 25.6%

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

   12. Applied egg-rr 25.6%

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

       1. *-lft-identity 25.6%

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

   14. Simplified 25.6%

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

   if -6.79999999999999998e-308 < z

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. sqr-neg 79.4%

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

      3. times-frac 85.2%

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

      4. sqr-neg 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in z around 0 63.8%

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

      1. mul-1-neg 63.8%

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

      2. unsub-neg 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in z around inf 12.6%

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

      1. mul-1-neg 12.6%

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

      2. associate-*l/ 13.3%

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

      3. *-commutative 13.3%

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

      4. distribute-rgt-neg-in 13.3%

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

      5. distribute-frac-neg2 13.3%

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

   10. Simplified 13.3%

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

       1. distribute-frac-neg2 13.3%

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

       2. distribute-frac-neg 13.3%

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

       3. associate-*r/ 12.6%

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

   12. Applied egg-rr 12.6%

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

       1. add-sqr-sqrt 12.6%

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

       2. times-frac 12.6%

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

       3. add-sqr-sqrt 5.7%

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

       4. sqrt-unprod 27.8%

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

       5. sqr-neg 27.8%

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

       6. sqrt-unprod 19.3%

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

       7. add-sqr-sqrt 32.5%

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

       8. times-frac 28.8%

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

       9. add-sqr-sqrt 28.9%

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

       10. clear-num 28.9%

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

       11. associate-/r* 36.2%

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

       12. associate-/r/ 36.9%

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

       13. clear-num 35.5%

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

   14. Applied egg-rr 35.5%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.8% accurate, 1.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ y_m (* z (/ 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 * (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 * (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 * (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 * (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 * 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 * (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 * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-/l* 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-/r* 84.8%

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

   5. sqr-neg 84.8%

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

3. Simplified 84.8%

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

   1. associate-*r/ 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*r/ 86.2%

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

   4. associate-/r* 92.1%

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

   5. associate-*l/ 97.1%

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

6. Applied egg-rr 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-/l* 92.1%

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

8. Applied egg-rr 92.1%

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

   1. clear-num 91.7%

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

   2. un-div-inv 92.6%

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

   3. div-inv 92.6%

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

   4. clear-num 92.6%

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

10. Applied egg-rr 92.6%

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

11. Taylor expanded in z around 0 73.8%

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

Alternative 14: 80.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-/l* 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-/r* 84.8%

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

   5. sqr-neg 84.8%

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

3. Simplified 84.8%

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

   1. associate-*r/ 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*r/ 86.2%

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

   4. associate-/r* 92.1%

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

   5. associate-*l/ 97.1%

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

6. Applied egg-rr 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-/l* 92.1%

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

8. Applied egg-rr 92.1%

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

9. Taylor expanded in z around 0 72.2%

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

Alternative 15: 32.5% accurate, 2.2× speedup

Math

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

FPCore

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

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));
}

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))
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));
}

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))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(x_m / z) * y_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 * ((x_m / z) * y_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[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. sqr-neg 78.4%

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

   3. times-frac 87.5%

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

   4. sqr-neg 87.5%

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

3. Simplified 87.5%

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

5. Taylor expanded in z around 0 64.0%

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

   1. mul-1-neg 64.0%

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

   2. unsub-neg 64.0%

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

7. Simplified 64.0%

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

8. Taylor expanded in z around inf 26.7%

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

   1. mul-1-neg 26.7%

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

   2. associate-*l/ 29.3%

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

   3. *-commutative 29.3%

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

   4. distribute-rgt-neg-in 29.3%

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

   5. distribute-frac-neg2 29.3%

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

10. Simplified 29.3%

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

    1. add-sqr-sqrt 19.4%

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

    2. sqrt-unprod 39.3%

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

    3. distribute-frac-neg2 39.3%

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

    4. distribute-frac-neg2 39.3%

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

    5. sqr-neg 39.3%

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

    6. sqrt-unprod 22.0%

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

    7. add-sqr-sqrt 29.9%

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

    8. *-un-lft-identity 29.9%

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

12. Applied egg-rr 29.9%

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

    1. *-lft-identity 29.9%

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

14. Simplified 29.9%

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

15. Final simplification 29.9%

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

Developer Target 1: 96.7% 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 2024185 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

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

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