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

Percentage Accurate: 83.1% → 97.6%

Time: 14.5s

Alternatives: 14

Speedup: 1.0×

• 25.9% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-40} \lor \neg \left(z \leq 2 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z + 1}}{z \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -2.9e-40) (not (<= z 2e-107)))
     (/ (/ x_m (+ z 1.0)) (* z (/ z y_m)))
     (/ (/ y_m (/ z x_m)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -2.9e-40) || !(z <= 2e-107)) {
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.9d-40)) .or. (.not. (z <= 2d-107))) then
        tmp = (x_m / (z + 1.0d0)) / (z * (z / y_m))
    else
        tmp = (y_m / (z / x_m)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -2.9e-40) || !(z <= 2e-107)) {
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	} else {
		tmp = (y_m / (z / x_m)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -2.9e-40) or not (z <= 2e-107):
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m))
	else:
		tmp = (y_m / (z / x_m)) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -2.9e-40) || !(z <= 2e-107))
		tmp = Float64(Float64(x_m / Float64(z + 1.0)) / Float64(z * Float64(z / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z <= -2.9e-40) || ~((z <= 2e-107)))
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	else
		tmp = (y_m / (z / x_m)) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -2.9e-40], N[Not[LessEqual[z, 2e-107]], $MachinePrecision]], N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999999e-40 or 2e-107 < z

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. frac-times 92.7%

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

      3. associate-*l/ 92.5%

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

      4. times-frac 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. clear-num 98.4%

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

      2. frac-times 97.1%

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

      3. *-un-lft-identity 97.1%

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

   6. Applied egg-rr 97.1%

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

   if -2.8999999999999999e-40 < z < 2e-107

   1. Initial program 79.0%

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

      1. associate-*l/ 73.5%

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

      2. sqr-neg 73.5%

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

      3. *-commutative 73.5%

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

      4. distribute-rgt1-in 73.5%

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

      5. sqr-neg 73.5%

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

      6. sqr-neg 73.5%

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

      7. cube-unmult 73.5%

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

   3. Simplified 73.5%

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

      1. associate-*l/ 79.0%

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

      2. *-commutative 79.0%

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

      3. cube-mult 79.0%

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

      4. distribute-rgt1-in 79.0%

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

      5. *-commutative 79.0%

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

      6. frac-times 73.7%

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

      7. associate-*l/ 79.0%

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

      8. associate-/r* 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. associate-/l* 97.9%

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

      2. div-inv 97.8%

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

      3. associate-/r/ 97.8%

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

   8. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. *-commutative 97.9%

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

      4. +-commutative 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in z around 0 97.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-40} \lor \neg \left(z \leq 2 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{\frac{x}{z + 1}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 0.002:\\ \;\;\;\;\frac{\frac{x\_m}{z + 1}}{z \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\left(z + 1\right) \cdot \frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 0.002)
     (/ (/ x_m (+ z 1.0)) (* z (/ z y_m)))
     (/ (/ y_m (* (+ z 1.0) (/ z x_m))) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 0.002) {
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	} else {
		tmp = (y_m / ((z + 1.0) * (z / x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * y_m) / ((z * z) * (z + 1.0d0))) <= 0.002d0) then
        tmp = (x_m / (z + 1.0d0)) / (z * (z / y_m))
    else
        tmp = (y_m / ((z + 1.0d0) * (z / x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 0.002) {
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	} else {
		tmp = (y_m / ((z + 1.0) * (z / x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((x_m * y_m) / ((z * z) * (z + 1.0))) <= 0.002:
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m))
	else:
		tmp = (y_m / ((z + 1.0) * (z / x_m))) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 0.002)
		tmp = Float64(Float64(x_m / Float64(z + 1.0)) / Float64(z * Float64(z / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(Float64(z + 1.0) * Float64(z / x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 0.002)
		tmp = (x_m / (z + 1.0)) / (z * (z / y_m));
	else
		tmp = (y_m / ((z + 1.0) * (z / x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(N[(z + 1.0), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 2e-3

   1. Initial program 90.5%

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

      1. *-commutative 90.5%

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

      2. frac-times 92.5%

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

      3. associate-*l/ 93.4%

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

      4. times-frac 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. clear-num 97.9%

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

      2. frac-times 93.9%

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

      3. *-un-lft-identity 93.9%

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

   6. Applied egg-rr 93.9%

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

   if 2e-3 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 65.1%

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

      1. associate-*l/ 68.5%

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

      2. sqr-neg 68.5%

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

      3. *-commutative 68.5%

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

      4. distribute-rgt1-in 65.8%

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

      5. sqr-neg 65.8%

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

      6. sqr-neg 65.8%

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

      7. cube-unmult 65.9%

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

   3. Simplified 65.9%

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

      1. associate-*l/ 65.1%

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

      2. *-commutative 65.1%

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

      3. cube-mult 65.1%

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

      4. distribute-rgt1-in 65.1%

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

      5. *-commutative 65.1%

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

      6. frac-times 70.9%

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

      7. associate-*l/ 74.9%

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

      8. associate-/r* 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. associate-/l* 96.1%

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

      2. div-inv 95.9%

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

      3. associate-/r/ 95.9%

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

   8. Applied egg-rr 95.9%

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

      1. associate-*r/ 96.1%

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

      2. *-rgt-identity 96.1%

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

      3. *-commutative 96.1%

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

      4. +-commutative 96.1%

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

   10. Simplified 96.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 0.002:\\ \;\;\;\;\frac{\frac{x}{z + 1}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(z + 1\right) \cdot \frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\frac{z}{x\_m}}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x\_m \cdot \frac{-y\_m}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ y_m (/ z x_m))))
   (*
    y_s
    (*
     x_s
     (if (<= z -1.0)
       t_0
       (if (<= z -4e-310)
         (* x_m (/ (- y_m) z))
         (if (<= z 2e-35) (/ x_m (/ z y_m)) t_0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m / (z / x_m);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= -4e-310) {
		tmp = x_m * (-y_m / z);
	} else if (z <= 2e-35) {
		tmp = x_m / (z / y_m);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m / (z / x_m)
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= (-4d-310)) then
        tmp = x_m * (-y_m / z)
    else if (z <= 2d-35) then
        tmp = x_m / (z / y_m)
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m / (z / x_m);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= -4e-310) {
		tmp = x_m * (-y_m / z);
	} else if (z <= 2e-35) {
		tmp = x_m / (z / y_m);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m / (z / x_m)
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= -4e-310:
		tmp = x_m * (-y_m / z)
	elif z <= 2e-35:
		tmp = x_m / (z / y_m)
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m / Float64(z / x_m))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= -4e-310)
		tmp = Float64(x_m * Float64(Float64(-y_m) / z));
	elseif (z <= 2e-35)
		tmp = Float64(x_m / Float64(z / y_m));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m / (z / x_m);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= -4e-310)
		tmp = x_m * (-y_m / z);
	elseif (z <= 2e-35)
		tmp = x_m / (z / y_m);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, -4e-310], N[(x$95$m * N[((-y$95$m) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-35], N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 2.00000000000000002e-35 < z

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. frac-times 91.4%

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

      3. associate-*l/ 91.9%

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

      4. times-frac 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around 0 38.2%

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

      1. neg-mul-1 38.2%

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

      2. +-commutative 38.2%

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

      3. unsub-neg 38.2%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in z around inf 28.4%

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

      1. associate-*r/ 28.4%

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

      2. mul-1-neg 28.4%

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

      3. distribute-rgt-neg-out 28.4%

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

      4. associate-*l/ 33.2%

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

   10. Simplified 33.2%

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

       1. clear-num 33.2%

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

       2. associate-*l/ 33.2%

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

       3. *-un-lft-identity 33.2%

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

       4. add-sqr-sqrt 15.5%

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

       5. sqrt-unprod 33.6%

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

       6. sqr-neg 33.6%

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

       7. sqrt-unprod 19.3%

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

       8. add-sqr-sqrt 36.3%

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

   12. Applied egg-rr 36.3%

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

   if -1 < z < -3.999999999999988e-310

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. frac-times 78.5%

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

      3. associate-*l/ 87.1%

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

      4. times-frac 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in z around 0 92.9%

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

      1. neg-mul-1 92.9%

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

      2. +-commutative 92.9%

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

      3. unsub-neg 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in z around inf 40.3%

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

      1. neg-mul-1 40.3%

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

   10. Simplified 40.3%

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

   if -3.999999999999988e-310 < z < 2.00000000000000002e-35

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. frac-times 79.8%

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

      3. associate-*l/ 78.1%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 96.7%

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

      1. neg-mul-1 96.7%

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

      2. +-commutative 96.7%

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

      3. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in z around inf 0.8%

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

      1. associate-*r/ 0.8%

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

      2. mul-1-neg 0.8%

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

      3. distribute-rgt-neg-out 0.8%

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

      4. associate-*l/ 0.8%

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

   10. Simplified 0.8%

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

       1. associate-*l/ 0.8%

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

       2. associate-/l* 0.8%

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

       3. add-sqr-sqrt 0.5%

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

       4. sqrt-unprod 19.8%

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

       5. sqr-neg 19.8%

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

       6. sqrt-unprod 19.2%

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

       7. add-sqr-sqrt 45.7%

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

   12. Applied egg-rr 45.7%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ y_m z) (/ (/ x_m z) z))
     (/ (/ y_m (/ z x_m)) z)))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (y_m / z) * ((x_m / z) / z)
    else
        tmp = (y_m / (z / x_m)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

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

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (y_m / z) * ((x_m / z) / z)
	else:
		tmp = (y_m / (z / x_m)) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / z) / z));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. frac-times 91.6%

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

      3. associate-*l/ 91.4%

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

      4. times-frac 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf 96.4%

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

   if -1 < z < 1

   1. Initial program 83.7%

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

      1. associate-*l/ 79.6%

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

      2. sqr-neg 79.6%

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

      3. *-commutative 79.6%

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

      4. distribute-rgt1-in 79.6%

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

      5. sqr-neg 79.6%

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

      6. sqr-neg 79.6%

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

      7. cube-unmult 79.6%

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

   3. Simplified 79.6%

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

      1. associate-*l/ 83.6%

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

      2. *-commutative 83.6%

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

      3. cube-mult 83.6%

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

      4. distribute-rgt1-in 83.7%

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

      5. *-commutative 83.7%

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

      6. frac-times 79.7%

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

      7. associate-*l/ 83.7%

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

      8. associate-/r* 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. associate-/l* 97.6%

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

      2. div-inv 97.5%

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

      3. associate-/r/ 97.5%

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

   8. Applied egg-rr 97.5%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. *-commutative 97.6%

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

      4. +-commutative 97.6%

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

   10. Simplified 97.6%

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

   11. Taylor expanded in z around 0 95.1%

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

4. Final simplification 95.8%

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

Alternative 5: 95.4% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (/ (/ (* y_m (/ x_m z)) z) z)
     (/ (/ y_m (/ z x_m)) z)))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((y_m * (x_m / z)) / z) / z
    else
        tmp = (y_m / (z / x_m)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

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

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((y_m * (x_m / z)) / z) / z
	else:
		tmp = (y_m / (z / x_m)) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(y_m * Float64(x_m / z)) / z) / z);
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 81.0%

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

      1. associate-*l/ 87.0%

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

      2. sqr-neg 87.0%

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

      3. *-commutative 87.0%

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

      4. distribute-rgt1-in 67.8%

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

      5. sqr-neg 67.8%

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

      6. sqr-neg 67.8%

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

      7. cube-unmult 67.8%

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

   3. Simplified 67.8%

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

      1. associate-*l/ 63.5%

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

      2. *-commutative 63.5%

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

      3. cube-mult 63.5%

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

      4. distribute-rgt1-in 81.0%

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

      5. *-commutative 81.0%

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

      6. frac-times 91.6%

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

      7. associate-*l/ 91.4%

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

      8. associate-/r* 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in z around inf 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 95.1%

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

   9. Simplified 95.1%

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

   if -1 < z < 1

   1. Initial program 83.7%

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

      1. associate-*l/ 79.6%

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

      2. sqr-neg 79.6%

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

      3. *-commutative 79.6%

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

      4. distribute-rgt1-in 79.6%

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

      5. sqr-neg 79.6%

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

      6. sqr-neg 79.6%

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

      7. cube-unmult 79.6%

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

   3. Simplified 79.6%

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

      1. associate-*l/ 83.6%

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

      2. *-commutative 83.6%

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

      3. cube-mult 83.6%

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

      4. distribute-rgt1-in 83.7%

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

      5. *-commutative 83.7%

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

      6. frac-times 79.7%

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

      7. associate-*l/ 83.7%

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

      8. associate-/r* 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. associate-/l* 97.6%

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

      2. div-inv 97.5%

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

      3. associate-/r/ 97.5%

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

   8. Applied egg-rr 97.5%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. *-commutative 97.6%

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

      4. +-commutative 97.6%

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

   10. Simplified 97.6%

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

   11. Taylor expanded in z around 0 95.1%

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

4. Final simplification 95.1%

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

Alternative 6: 36.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310} \lor \neg \left(z \leq 10^{-37}\right):\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -4e-310) (not (<= z 1e-37)))
     (/ y_m (/ z x_m))
     (/ 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) {
	double tmp;
	if ((z <= -4e-310) || !(z <= 1e-37)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m / (z / y_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 <= (-4d-310)) .or. (.not. (z <= 1d-37))) then
        tmp = y_m / (z / x_m)
    else
        tmp = x_m / (z / y_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 <= -4e-310) || !(z <= 1e-37)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m / (z / y_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 <= -4e-310) or not (z <= 1e-37):
		tmp = y_m / (z / x_m)
	else:
		tmp = x_m / (z / y_m)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -4e-310) || !(z <= 1e-37))
		tmp = Float64(y_m / Float64(z / x_m));
	else
		tmp = Float64(x_m / Float64(z / y_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 <= -4e-310) || ~((z <= 1e-37)))
		tmp = y_m / (z / x_m);
	else
		tmp = x_m / (z / y_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, -4e-310], N[Not[LessEqual[z, 1e-37]], $MachinePrecision]], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.999999999999988e-310 or 1.00000000000000007e-37 < z

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. frac-times 87.4%

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

      3. associate-*l/ 90.4%

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

      4. times-frac 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in z around 0 55.5%

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

      1. neg-mul-1 55.5%

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

      2. +-commutative 55.5%

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

      3. unsub-neg 55.5%

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

   7. Simplified 55.5%

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

   8. Taylor expanded in z around inf 29.0%

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

      1. associate-*r/ 29.0%

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

      2. mul-1-neg 29.0%

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

      3. distribute-rgt-neg-out 29.0%

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

      4. associate-*l/ 35.2%

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

   10. Simplified 35.2%

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

       1. clear-num 35.2%

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

       2. associate-*l/ 35.2%

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

       3. *-un-lft-identity 35.2%

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

       4. add-sqr-sqrt 15.7%

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

       5. sqrt-unprod 28.1%

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

       6. sqr-neg 28.1%

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

       7. sqrt-unprod 13.3%

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

       8. add-sqr-sqrt 25.5%

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

   12. Applied egg-rr 25.5%

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

   if -3.999999999999988e-310 < z < 1.00000000000000007e-37

   1. Initial program 77.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. 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. frac-times 79.5%

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

      3. associate-*l/ 77.8%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 96.7%

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

      1. neg-mul-1 96.7%

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

      2. +-commutative 96.7%

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

      3. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in z around inf 0.8%

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

      1. associate-*r/ 0.8%

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

      2. mul-1-neg 0.8%

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

      3. distribute-rgt-neg-out 0.8%

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

      4. associate-*l/ 0.7%

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

   10. Simplified 0.7%

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

       1. associate-*l/ 0.8%

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

       2. associate-/l* 0.7%

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

       3. add-sqr-sqrt 0.5%

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

       4. sqrt-unprod 20.1%

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

       5. sqr-neg 20.1%

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

       6. sqrt-unprod 19.6%

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

       7. add-sqr-sqrt 46.3%

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

   12. Applied egg-rr 46.3%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310} \lor \neg \left(z \leq 10^{-37}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;y\_m \cdot \frac{-x\_m}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -4e-310)
     (* y_m (/ (- x_m) z))
     (if (<= z 2e-35) (/ x_m (/ z y_m)) (/ y_m (/ z x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -4e-310) {
		tmp = y_m * (-x_m / z);
	} else if (z <= 2e-35) {
		tmp = x_m / (z / y_m);
	} else {
		tmp = y_m / (z / x_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4d-310)) then
        tmp = y_m * (-x_m / z)
    else if (z <= 2d-35) then
        tmp = x_m / (z / y_m)
    else
        tmp = y_m / (z / x_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -4e-310) {
		tmp = y_m * (-x_m / z);
	} else if (z <= 2e-35) {
		tmp = x_m / (z / y_m);
	} else {
		tmp = y_m / (z / x_m);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= -4e-310:
		tmp = y_m * (-x_m / z)
	elif z <= 2e-35:
		tmp = x_m / (z / y_m)
	else:
		tmp = y_m / (z / x_m)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -4e-310)
		tmp = Float64(y_m * Float64(Float64(-x_m) / z));
	elseif (z <= 2e-35)
		tmp = Float64(x_m / Float64(z / y_m));
	else
		tmp = Float64(y_m / Float64(z / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -4e-310)
		tmp = y_m * (-x_m / z);
	elseif (z <= 2e-35)
		tmp = x_m / (z / y_m);
	else
		tmp = y_m / (z / x_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -4e-310], N[(y$95$m * N[((-x$95$m) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-35], N[(x$95$m / N[(z / y$95$m), $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 < -3.999999999999988e-310

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. frac-times 85.3%

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

      3. associate-*l/ 89.0%

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

      4. times-frac 95.2%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in z around 0 66.2%

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

      1. neg-mul-1 66.2%

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

      2. +-commutative 66.2%

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

      3. unsub-neg 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in z around inf 30.0%

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

      1. associate-*r/ 30.0%

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

      2. mul-1-neg 30.0%

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

      3. distribute-rgt-neg-out 30.0%

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

      4. associate-*l/ 35.8%

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

   10. Simplified 35.8%

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

   if -3.999999999999988e-310 < z < 2.00000000000000002e-35

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. frac-times 79.8%

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

      3. associate-*l/ 78.1%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 96.7%

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

      1. neg-mul-1 96.7%

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

      2. +-commutative 96.7%

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

      3. unsub-neg 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in z around inf 0.8%

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

      1. associate-*r/ 0.8%

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

      2. mul-1-neg 0.8%

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

      3. distribute-rgt-neg-out 0.8%

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

      4. associate-*l/ 0.8%

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

   10. Simplified 0.8%

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

       1. associate-*l/ 0.8%

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

       2. associate-/l* 0.8%

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

       3. add-sqr-sqrt 0.5%

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

       4. sqrt-unprod 19.8%

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

       5. sqr-neg 19.8%

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

       6. sqrt-unprod 19.2%

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

       7. add-sqr-sqrt 45.7%

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

   12. Applied egg-rr 45.7%

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

   if 2.00000000000000002e-35 < z

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. frac-times 90.3%

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

      3. associate-*l/ 92.3%

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

      4. times-frac 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in z around 0 39.6%

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

      1. neg-mul-1 39.6%

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

      2. +-commutative 39.6%

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

      3. unsub-neg 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in z around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. mul-1-neg 27.8%

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

      3. distribute-rgt-neg-out 27.8%

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

      4. associate-*l/ 34.8%

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

   10. Simplified 34.8%

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

       1. clear-num 34.8%

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

       2. associate-*l/ 34.8%

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

       3. *-un-lft-identity 34.8%

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

       4. add-sqr-sqrt 15.3%

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

       5. sqrt-unprod 31.8%

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

       6. sqr-neg 31.8%

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

       7. sqrt-unprod 21.6%

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

       8. add-sqr-sqrt 37.9%

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

   12. Applied egg-rr 37.9%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 7.5e-82) (* (/ y_m z) (/ x_m z)) (* x_m (/ y_m (* z z)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 7.5e-82) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 7.5d-82) then
        tmp = (y_m / z) * (x_m / z)
    else
        tmp = x_m * (y_m / (z * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 7.5e-82) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = x_m * (y_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 7.5e-82:
		tmp = (y_m / z) * (x_m / z)
	else:
		tmp = x_m * (y_m / (z * z))
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 7.5e-82)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 7.5e-82)
		tmp = (y_m / z) * (x_m / z);
	else
		tmp = x_m * (y_m / (z * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 7.5e-82], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999997e-82

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. frac-times 83.7%

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

      3. associate-*l/ 86.1%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 74.5%

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

   if 7.4999999999999997e-82 < x

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. times-frac 90.3%

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

      4. sqr-neg 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around 0 72.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+77}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= y_m 1e+77) (* (/ y_m z) (/ x_m z)) (* y_m (/ (/ x_m z) z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+77) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = y_m * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1d+77) then
        tmp = (y_m / z) * (x_m / z)
    else
        tmp = y_m * ((x_m / z) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+77) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = y_m * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 1e+77:
		tmp = (y_m / z) * (x_m / z)
	else:
		tmp = y_m * ((x_m / z) / z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+77)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(x_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 1e+77)
		tmp = (y_m / z) * (x_m / z);
	else
		tmp = y_m * ((x_m / z) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e+77], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999983e76

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. frac-times 86.3%

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

      3. associate-*l/ 87.4%

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

      4. times-frac 97.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in z around 0 75.1%

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

   if 9.99999999999999983e76 < y

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. frac-times 82.9%

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

      3. associate-*l/ 87.8%

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

      4. times-frac 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in z around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. clear-num 56.7%

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

      3. frac-times 70.3%

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

      4. *-un-lft-identity 70.3%

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

   7. Applied egg-rr 70.3%

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

      1. clear-num 70.2%

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

      2. associate-/r/ 70.3%

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

      3. associate-/r* 70.1%

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

      4. clear-num 70.1%

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

   9. Applied egg-rr 70.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+77}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-58}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= y_m 1e-58) (* (/ y_m z) (/ x_m 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 (y_m <= 1e-58) {
		tmp = (y_m / z) * (x_m / 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 (y_m <= 1d-58) then
        tmp = (y_m / z) * (x_m / 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 (y_m <= 1e-58) {
		tmp = (y_m / z) * (x_m / 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 y_m <= 1e-58:
		tmp = (y_m / z) * (x_m / 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 (y_m <= 1e-58)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / 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 (y_m <= 1e-58)
		tmp = (y_m / z) * (x_m / 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[LessEqual[y$95$m, 1e-58], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / 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 y < 1e-58

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. frac-times 84.6%

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

      3. associate-*l/ 85.9%

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

      4. times-frac 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in z around 0 74.8%

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

   if 1e-58 < y

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. frac-times 87.7%

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

      3. associate-*l/ 91.2%

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

      4. times-frac 95.7%

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

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in z around 0 62.9%

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

      1. *-commutative 62.9%

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

      2. clear-num 62.8%

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

      3. frac-times 72.4%

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

      4. *-un-lft-identity 72.4%

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

   7. Applied egg-rr 72.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-58}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 5.6e-73) (/ (/ y_m z) (/ z x_m)) (/ 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 (y_m <= 5.6e-73) {
		tmp = (y_m / z) / (z / x_m);
	} 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 (y_m <= 5.6d-73) then
        tmp = (y_m / z) / (z / x_m)
    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 (y_m <= 5.6e-73) {
		tmp = (y_m / z) / (z / x_m);
	} 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 y_m <= 5.6e-73:
		tmp = (y_m / z) / (z / x_m)
	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 (y_m <= 5.6e-73)
		tmp = Float64(Float64(y_m / z) / Float64(z / x_m));
	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 (y_m <= 5.6e-73)
		tmp = (y_m / z) / (z / x_m);
	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[LessEqual[y$95$m, 5.6e-73], N[(N[(y$95$m / z), $MachinePrecision] / N[(z / x$95$m), $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 y < 5.60000000000000023e-73

   1. Initial program 81.5%

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

      1. associate-*l/ 81.0%

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

      2. sqr-neg 81.0%

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

      3. *-commutative 81.0%

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

      4. distribute-rgt1-in 71.8%

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

      5. sqr-neg 71.8%

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

      6. sqr-neg 71.8%

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

      7. cube-unmult 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*l/ 72.3%

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

      2. cube-mult 72.3%

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

      3. distribute-rgt1-in 81.5%

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

      4. frac-times 84.3%

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

      5. clear-num 84.3%

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

      6. associate-/r* 89.8%

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

      7. frac-times 97.2%

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

      8. *-un-lft-identity 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in z around 0 75.2%

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

   if 5.60000000000000023e-73 < y

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 88.3%

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

      3. associate-*l/ 91.6%

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

      4. times-frac 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in z around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. clear-num 62.5%

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

      3. frac-times 71.6%

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

      4. *-un-lft-identity 71.6%

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

   7. Applied egg-rr 71.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. *-commutative 82.3%

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

   2. frac-times 85.6%

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

   3. associate-*l/ 87.5%

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

   4. times-frac 96.8%

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

4. Applied egg-rr 96.8%

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

5. Final simplification 96.8%

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

Alternative 13: 74.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. *-commutative 82.3%

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

   2. frac-times 85.6%

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

   3. associate-*l/ 87.5%

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

   4. times-frac 96.8%

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

4. Applied egg-rr 96.8%

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

5. Taylor expanded in z around 0 71.2%

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

6. Final simplification 71.2%

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

Alternative 14: 28.8% accurate, 2.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 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(x_m / Float64(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[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. *-commutative 82.3%

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

   2. frac-times 85.6%

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

   3. associate-*l/ 87.5%

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

   4. times-frac 96.8%

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

4. Applied egg-rr 96.8%

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

5. Taylor expanded in z around 0 65.2%

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

   1. neg-mul-1 65.2%

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

   2. +-commutative 65.2%

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

   3. unsub-neg 65.2%

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

7. Simplified 65.2%

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

8. Taylor expanded in z around inf 22.4%

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

   1. associate-*r/ 22.4%

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

   2. mul-1-neg 22.4%

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

   3. distribute-rgt-neg-out 22.4%

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

   4. associate-*l/ 27.1%

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

10. Simplified 27.1%

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

    1. associate-*l/ 22.4%

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

    2. associate-/l* 27.2%

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

    3. add-sqr-sqrt 12.8%

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

    4. sqrt-unprod 28.0%

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

    5. sqr-neg 28.0%

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

    6. sqrt-unprod 14.4%

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

    7. add-sqr-sqrt 30.8%

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

12. Applied egg-rr 30.8%

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

13. Final simplification 30.8%

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

Developer target: 96.9% 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 2024031 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

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

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