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

Percentage Accurate: 83.3% → 97.6%

Time: 13.7s

Alternatives: 13

Speedup: 1.0×

• 26.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z \cdot \frac{z}{y\_m}}}{z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{x\_m}{z + 1} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ (/ x_m (* z (/ z y_m))) z)))
   (*
    y_s
    (*
     x_s
     (if (<= z -9e+69)
       t_0
       (if (<= z -1.15e-16)
         (* (/ x_m (+ z 1.0)) (/ y_m (* z z)))
         (if (<= z 0.46) (* y_m (/ (/ x_m z) z)) t_0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z * (z / y_m))) / z;
	double tmp;
	if (z <= -9e+69) {
		tmp = t_0;
	} else if (z <= -1.15e-16) {
		tmp = (x_m / (z + 1.0)) * (y_m / (z * z));
	} else if (z <= 0.46) {
		tmp = y_m * ((x_m / z) / z);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / (z * (z / y_m))) / z
    if (z <= (-9d+69)) then
        tmp = t_0
    else if (z <= (-1.15d-16)) then
        tmp = (x_m / (z + 1.0d0)) * (y_m / (z * z))
    else if (z <= 0.46d0) then
        tmp = y_m * ((x_m / z) / z)
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z * (z / y_m))) / z;
	double tmp;
	if (z <= -9e+69) {
		tmp = t_0;
	} else if (z <= -1.15e-16) {
		tmp = (x_m / (z + 1.0)) * (y_m / (z * z));
	} else if (z <= 0.46) {
		tmp = y_m * ((x_m / z) / z);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (x_m / (z * (z / y_m))) / z
	tmp = 0
	if z <= -9e+69:
		tmp = t_0
	elif z <= -1.15e-16:
		tmp = (x_m / (z + 1.0)) * (y_m / (z * z))
	elif z <= 0.46:
		tmp = y_m * ((x_m / z) / z)
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / Float64(z * Float64(z / y_m))) / z)
	tmp = 0.0
	if (z <= -9e+69)
		tmp = t_0;
	elseif (z <= -1.15e-16)
		tmp = Float64(Float64(x_m / Float64(z + 1.0)) * Float64(y_m / Float64(z * z)));
	elseif (z <= 0.46)
		tmp = Float64(y_m * Float64(Float64(x_m / z) / z));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (x_m / (z * (z / y_m))) / z;
	tmp = 0.0;
	if (z <= -9e+69)
		tmp = t_0;
	elseif (z <= -1.15e-16)
		tmp = (x_m / (z + 1.0)) * (y_m / (z * z));
	elseif (z <= 0.46)
		tmp = y_m * ((x_m / z) / z);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -9e+69], t$95$0, If[LessEqual[z, -1.15e-16], N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.46], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -8.9999999999999999e69 or 0.46000000000000002 < z

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/l* 81.3%

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

      3. sqr-neg 81.3%

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

      4. associate-/r* 85.6%

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

      5. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r/ 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*r/ 84.8%

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

      4. associate-/r* 94.1%

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

      5. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. clear-num 96.4%

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

      3. frac-times 95.6%

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

      4. *-un-lft-identity 95.6%

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

   9. Applied egg-rr 95.6%

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

   if -8.9999999999999999e69 < z < -1.15e-16

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. sqr-neg 92.2%

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

      3. times-frac 98.7%

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

      4. sqr-neg 98.7%

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

   3. Simplified 98.7%

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

   if -1.15e-16 < z < 0.46000000000000002

   1. Initial program 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-/l* 88.7%

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

      3. sqr-neg 88.7%

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

      4. associate-/r* 88.7%

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

      5. sqr-neg 88.7%

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

   3. Simplified 88.7%

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

      1. associate-*r/ 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*r/ 88.7%

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

      4. associate-/r* 93.4%

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

      5. associate-*l/ 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in z around 0 87.7%

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

      1. associate-*r/ 96.5%

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

   9. Simplified 96.5%

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

       1. associate-*l/ 96.5%

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

       2. *-commutative 96.5%

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

       3. div-inv 96.5%

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

       4. associate-*l* 92.6%

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

   11. Applied egg-rr 92.6%

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

       1. associate-*l/ 92.7%

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

       2. *-un-lft-identity 92.7%

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

   13. Applied egg-rr 92.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{y\_m}}{z}\\ y\_s \cdot \left(x\_s \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{x\_m}{z + 1}\right)\right)\right) \end{array} \end{array} \]

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 (/ (sqrt y_m) z)))
   (* y_s (* x_s (* t_0 (* t_0 (/ x_m (+ z 1.0))))))))

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 = sqrt(y_m) / z;
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
}

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
    t_0 = sqrt(y_m) / z
    code = y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0d0)))))
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 = Math.sqrt(y_m) / z;
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
}

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 = math.sqrt(y_m) / z
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))))

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(sqrt(y_m) / z)
	return Float64(y_s * Float64(x_s * Float64(t_0 * Float64(t_0 * Float64(x_m / Float64(z + 1.0))))))
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)
	t_0 = sqrt(y_m) / z;
	tmp = y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
end

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. frac-times 88.8%

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

   3. add-sqr-sqrt 59.1%

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

   4. associate-*l* 59.1%

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

   5. sqrt-div 45.7%

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

   6. sqrt-prod 17.5%

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

   7. add-sqr-sqrt 26.9%

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

   8. sqrt-div 28.0%

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

   9. sqrt-prod 20.4%

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

   10. add-sqr-sqrt 50.4%

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

4. Applied egg-rr 50.4%

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

5. Final simplification 50.4%

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

Alternative 3: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z \cdot \frac{z}{y\_m}}}{z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-110}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ (/ x_m (* z (/ z y_m))) z)))
   (*
    y_s
    (*
     x_s
     (if (<= z -5.2e+67)
       t_0
       (if (<= z -2e-110)
         (* y_m (/ (/ x_m (* z z)) (+ z 1.0)))
         (if (<= z 0.46) (* y_m (/ (/ x_m z) z)) t_0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z * (z / y_m))) / z;
	double tmp;
	if (z <= -5.2e+67) {
		tmp = t_0;
	} else if (z <= -2e-110) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 0.46) {
		tmp = y_m * ((x_m / z) / z);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / (z * (z / y_m))) / z
    if (z <= (-5.2d+67)) then
        tmp = t_0
    else if (z <= (-2d-110)) then
        tmp = y_m * ((x_m / (z * z)) / (z + 1.0d0))
    else if (z <= 0.46d0) then
        tmp = y_m * ((x_m / z) / z)
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z * (z / y_m))) / z;
	double tmp;
	if (z <= -5.2e+67) {
		tmp = t_0;
	} else if (z <= -2e-110) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 0.46) {
		tmp = y_m * ((x_m / z) / z);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (x_m / (z * (z / y_m))) / z
	tmp = 0
	if z <= -5.2e+67:
		tmp = t_0
	elif z <= -2e-110:
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0))
	elif z <= 0.46:
		tmp = y_m * ((x_m / z) / z)
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / Float64(z * Float64(z / y_m))) / z)
	tmp = 0.0
	if (z <= -5.2e+67)
		tmp = t_0;
	elseif (z <= -2e-110)
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / Float64(z + 1.0)));
	elseif (z <= 0.46)
		tmp = Float64(y_m * Float64(Float64(x_m / z) / z));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (x_m / (z * (z / y_m))) / z;
	tmp = 0.0;
	if (z <= -5.2e+67)
		tmp = t_0;
	elseif (z <= -2e-110)
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	elseif (z <= 0.46)
		tmp = y_m * ((x_m / z) / z);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -5.2e+67], t$95$0, If[LessEqual[z, -2e-110], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.46], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2000000000000001e67 or 0.46000000000000002 < z

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-/l* 81.1%

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

      3. sqr-neg 81.1%

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

      4. associate-/r* 85.3%

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

      5. sqr-neg 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-*r/ 85.1%

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

      4. associate-/r* 94.2%

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

      5. associate-*l/ 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in z around inf 96.4%

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

      1. *-commutative 96.4%

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

      2. clear-num 96.5%

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

      3. frac-times 95.7%

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

      4. *-un-lft-identity 95.7%

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

   9. Applied egg-rr 95.7%

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

   if -5.2000000000000001e67 < z < -2.0000000000000001e-110

   1. Initial program 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-/l* 99.4%

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

      3. sqr-neg 99.4%

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

      4. associate-/r* 99.7%

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

      5. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   if -2.0000000000000001e-110 < z < 0.46000000000000002

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-/l* 86.4%

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

      3. sqr-neg 86.4%

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

      4. associate-/r* 86.4%

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

      5. sqr-neg 86.4%

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

   3. Simplified 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-*r/ 86.4%

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

      4. associate-/r* 92.1%

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

      5. associate-*l/ 96.3%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in z around 0 87.1%

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

      1. associate-*r/ 97.2%

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

   9. Simplified 97.2%

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

       1. associate-*l/ 96.9%

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

       2. *-commutative 96.9%

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

       3. div-inv 96.8%

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

       4. associate-*l* 91.2%

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

   11. Applied egg-rr 91.2%

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

       1. associate-*l/ 91.2%

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

       2. *-un-lft-identity 91.2%

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

   13. Applied egg-rr 91.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z}}{z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ (/ x_m z) z)))
   (*
    y_s
    (*
     x_s
     (if (or (<= z -1.0) (not (<= z 0.46))) (* t_0 (/ y_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 = (x_m / z) / z;
	double tmp;
	if ((z <= -1.0) || !(z <= 0.46)) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / z) / z
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = t_0 * (y_m / z)
    else
        tmp = y_m * t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / z) / z;
	double tmp;
	if ((z <= -1.0) || !(z <= 0.46)) {
		tmp = t_0 * (y_m / z);
	} else {
		tmp = y_m * t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (x_m / z) / z
	tmp = 0
	if (z <= -1.0) or not (z <= 0.46):
		tmp = t_0 * (y_m / z)
	else:
		tmp = y_m * t_0
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(t_0 * Float64(y_m / z));
	else
		tmp = Float64(y_m * t_0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (x_m / z) / z;
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.46)))
		tmp = t_0 * (y_m / z);
	else
		tmp = y_m * 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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-/l* 83.7%

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

      3. sqr-neg 83.7%

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

      4. associate-/r* 87.4%

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

      5. sqr-neg 87.4%

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

   3. Simplified 87.4%

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

      1. associate-*r/ 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-*r/ 87.1%

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

      4. associate-/r* 94.9%

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

      5. associate-*l/ 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-/l* 91.7%

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

   9. Applied egg-rr 91.7%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/l* 89.1%

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

      3. sqr-neg 89.1%

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

      4. associate-/r* 89.2%

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

      5. sqr-neg 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r/ 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*r/ 89.2%

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

      4. associate-/r* 93.7%

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

      5. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*l/ 94.2%

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

       2. *-commutative 94.2%

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

       3. div-inv 94.2%

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

       4. associate-*l* 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. associate-*l/ 90.5%

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

       2. *-un-lft-identity 90.5%

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

   13. Applied egg-rr 90.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.0) (not (<= z 0.46)))
     (* (/ x_m z) (/ (/ y_m z) 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / z) * ((y_m / z) / 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 ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = (x_m / z) * ((y_m / z) / 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / z) * ((y_m / z) / 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 (z <= -1.0) or not (z <= 0.46):
		tmp = (x_m / z) * ((y_m / z) / 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 ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / 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 ((z <= -1.0) || ~((z <= 0.46)))
		tmp = (x_m / z) * ((y_m / z) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / 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 z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-/l* 83.7%

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

      3. sqr-neg 83.7%

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

      4. associate-/r* 87.4%

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

      5. sqr-neg 87.4%

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

   3. Simplified 87.4%

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

      1. associate-/l/ 83.7%

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

      2. associate-/r* 87.4%

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

      3. associate-/l* 88.4%

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

      4. associate-/r* 95.5%

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

      5. clear-num 94.8%

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

      6. un-div-inv 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-/l/ 87.9%

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

      2. associate-/r/ 89.8%

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

      3. times-frac 94.3%

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

   8. Applied egg-rr 94.3%

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

   9. Taylor expanded in z around inf 91.3%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/l* 89.1%

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

      3. sqr-neg 89.1%

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

      4. associate-/r* 89.2%

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

      5. sqr-neg 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r/ 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*r/ 89.2%

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

      4. associate-/r* 93.7%

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

      5. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*l/ 94.2%

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

       2. *-commutative 94.2%

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

       3. div-inv 94.2%

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

       4. associate-*l* 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. associate-*l/ 90.5%

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

       2. *-un-lft-identity 90.5%

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

   13. Applied egg-rr 90.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{x\_m}{z \cdot \frac{z}{y\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.0) (not (<= z 0.46)))
     (/ (/ x_m (* z (/ z y_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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / (z * (z / y_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 ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = (x_m / (z * (z / y_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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / (z * (z / y_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 (z <= -1.0) or not (z <= 0.46):
		tmp = (x_m / (z * (z / y_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 ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(Float64(x_m / Float64(z * Float64(z / y_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 ((z <= -1.0) || ~((z <= 0.46)))
		tmp = (x_m / (z * (z / y_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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(N[(x$95$m / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $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 z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-/l* 83.7%

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

      3. sqr-neg 83.7%

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

      4. associate-/r* 87.4%

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

      5. sqr-neg 87.4%

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

   3. Simplified 87.4%

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

      1. associate-*r/ 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-*r/ 87.1%

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

      4. associate-/r* 94.9%

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

      5. associate-*l/ 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 93.0%

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

      1. *-commutative 93.0%

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

      2. clear-num 92.8%

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

      3. frac-times 92.1%

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

      4. *-un-lft-identity 92.1%

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

   9. Applied egg-rr 92.1%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/l* 89.1%

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

      3. sqr-neg 89.1%

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

      4. associate-/r* 89.2%

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

      5. sqr-neg 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r/ 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*r/ 89.2%

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

      4. associate-/r* 93.7%

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

      5. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*l/ 94.2%

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

       2. *-commutative 94.2%

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

       3. div-inv 94.2%

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

       4. associate-*l* 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. associate-*l/ 90.5%

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

       2. *-un-lft-identity 90.5%

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

   13. Applied egg-rr 90.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z}}{z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y\_m \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ x_m z) z)))
   (*
    y_s
    (*
     x_s
     (if (<= z -1.0)
       (* t_0 (/ y_m z))
       (if (<= z 0.46) (* y_m t_0) (* (/ x_m z) (/ y_m (* z z)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / z) / z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0 * (y_m / z);
	} else if (z <= 0.46) {
		tmp = y_m * t_0;
	} else {
		tmp = (x_m / z) * (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) :: t_0
    real(8) :: tmp
    t_0 = (x_m / z) / z
    if (z <= (-1.0d0)) then
        tmp = t_0 * (y_m / z)
    else if (z <= 0.46d0) then
        tmp = y_m * t_0
    else
        tmp = (x_m / z) * (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 t_0 = (x_m / z) / z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0 * (y_m / z);
	} else if (z <= 0.46) {
		tmp = y_m * t_0;
	} else {
		tmp = (x_m / z) * (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):
	t_0 = (x_m / z) / z
	tmp = 0
	if z <= -1.0:
		tmp = t_0 * (y_m / z)
	elif z <= 0.46:
		tmp = y_m * t_0
	else:
		tmp = (x_m / z) * (y_m / (z * z))
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(t_0 * Float64(y_m / z));
	elseif (z <= 0.46)
		tmp = Float64(y_m * t_0);
	else
		tmp = Float64(Float64(x_m / z) * 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)
	t_0 = (x_m / z) / z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0 * (y_m / z);
	elseif (z <= 0.46)
		tmp = y_m * t_0;
	else
		tmp = (x_m / z) * (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_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.46], N[(y$95$m * t$95$0), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   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. *-commutative 81.5%

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

      2. associate-/l* 85.7%

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

      3. sqr-neg 85.7%

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

      4. associate-/r* 89.2%

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

      5. sqr-neg 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r/ 88.8%

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

      2. *-commutative 88.8%

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

      3. associate-*r/ 88.8%

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

      4. associate-/r* 96.5%

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

      5. associate-*l/ 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in z around inf 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-/l* 91.6%

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

   9. Applied egg-rr 91.6%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/l* 89.1%

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

      3. sqr-neg 89.1%

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

      4. associate-/r* 89.2%

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

      5. sqr-neg 89.2%

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

   3. Simplified 89.2%

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

      1. associate-*r/ 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*r/ 89.2%

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

      4. associate-/r* 93.7%

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

      5. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*l/ 94.2%

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

       2. *-commutative 94.2%

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

       3. div-inv 94.2%

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

       4. associate-*l* 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. associate-*l/ 90.5%

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

       2. *-un-lft-identity 90.5%

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

   13. Applied egg-rr 90.5%

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

   if 0.46000000000000002 < z

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. sqr-neg 83.5%

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

      3. times-frac 85.9%

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

      4. sqr-neg 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in z around inf 85.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z -5e-310)))
     (/ y_m (/ z x_m))
     (* x_m (/ y_m (- z)))))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= (-5d-310)))) then
        tmp = y_m / (z / x_m)
    else
        tmp = x_m * (y_m / -z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= -5e-310)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m * (y_m / -z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= -5e-310):
		tmp = y_m / (z / x_m)
	else:
		tmp = x_m * (y_m / -z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= -5e-310))
		tmp = Float64(y_m / Float64(z / x_m));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(-z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= -5e-310)))
		tmp = y_m / (z / x_m);
	else
		tmp = x_m * (y_m / -z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, -5e-310]], $MachinePrecision]], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/l* 84.3%

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

      3. sqr-neg 84.3%

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

      4. associate-/r* 86.8%

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

      5. sqr-neg 86.8%

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

   3. Simplified 86.8%

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

      1. associate-*r/ 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*r/ 86.6%

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

      4. associate-/r* 93.9%

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

      5. associate-*l/ 95.7%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in z around 0 42.0%

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

      1. neg-mul-1 42.0%

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

      2. distribute-rgt-neg-in 42.0%

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

      3. associate-*r/ 45.5%

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

      4. distribute-lft-out 50.3%

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

   9. Simplified 50.3%

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

   10. Taylor expanded in z around inf 20.5%

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

       1. mul-1-neg 20.5%

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

       2. distribute-frac-neg2 20.5%

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

       3. associate-/l* 25.0%

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

   12. Simplified 25.0%

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

       1. associate-*r/ 20.5%

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

       2. add-sqr-sqrt 11.6%

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

       3. *-commutative 11.6%

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

       4. sqrt-unprod 49.7%

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

       5. sqr-neg 49.7%

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

       6. sqrt-unprod 19.0%

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

       7. add-sqr-sqrt 31.4%

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

       8. associate-*l/ 36.4%

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

       9. associate-/r/ 38.8%

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

   14. Applied egg-rr 38.8%

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

   if -1 < z < -4.999999999999985e-310

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/l* 92.5%

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

      3. sqr-neg 92.5%

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

      4. associate-/r* 92.6%

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

      5. sqr-neg 92.6%

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

   3. Simplified 92.6%

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

      1. associate-*r/ 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*r/ 92.5%

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

      4. associate-/r* 95.5%

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

      5. associate-*l/ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in z around 0 88.1%

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

      1. neg-mul-1 88.1%

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

      2. distribute-rgt-neg-in 88.1%

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

      3. associate-*r/ 94.0%

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

      4. distribute-lft-out 94.0%

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

   9. Simplified 94.0%

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

   10. Taylor expanded in z around inf 47.0%

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

       1. mul-1-neg 47.0%

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

       2. distribute-frac-neg2 47.0%

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

       3. associate-/l* 54.2%

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

   12. Simplified 54.2%

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

4. Final simplification 42.7%

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

Alternative 9: 97.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 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 1.0)) z) (/ x_m z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-/l* 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-/r* 88.3%

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

   5. sqr-neg 88.3%

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

3. Simplified 88.3%

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

   1. associate-/l/ 86.4%

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

   2. associate-/r* 88.2%

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

   3. associate-/l* 87.9%

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

   4. associate-/r* 92.3%

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

   5. clear-num 92.0%

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

   6. un-div-inv 92.0%

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

6. Applied egg-rr 92.0%

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

   1. associate-/l/ 87.6%

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

   2. associate-/r/ 88.6%

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

   3. times-frac 95.8%

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

8. Applied egg-rr 95.8%

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

9. Final simplification 95.8%

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

Alternative 10: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-/l* 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-/r* 88.3%

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

   5. sqr-neg 88.3%

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

3. Simplified 88.3%

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

   1. associate-*r/ 88.1%

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

   2. *-commutative 88.1%

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

   3. associate-*r/ 88.1%

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

   4. associate-/r* 94.3%

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

   5. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

   1. *-commutative 96.4%

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

   2. clear-num 96.2%

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

   3. frac-times 96.3%

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

   4. *-un-lft-identity 96.3%

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

8. Applied egg-rr 96.3%

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

9. Final simplification 96.3%

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

Alternative 11: 80.5% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 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 (/ (/ x_m z) z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-/l* 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-/r* 88.3%

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

   5. sqr-neg 88.3%

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

3. Simplified 88.3%

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

   1. associate-*r/ 88.1%

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

   2. *-commutative 88.1%

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

   3. associate-*r/ 88.1%

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

   4. associate-/r* 94.3%

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

   5. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

7. Taylor expanded in z around 0 65.6%

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

   1. associate-*r/ 70.8%

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

9. Simplified 70.8%

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

    1. associate-*l/ 72.0%

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

    2. *-commutative 72.0%

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

    3. div-inv 72.0%

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

    4. associate-*l* 72.0%

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

11. Applied egg-rr 72.0%

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

    1. associate-*l/ 72.0%

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

    2. *-un-lft-identity 72.0%

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

13. Applied egg-rr 72.0%

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

14. Final simplification 72.0%

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

Alternative 12: 28.4% accurate, 2.2× speedup

Math

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

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 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-/l* 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-/r* 88.3%

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

   5. sqr-neg 88.3%

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

3. Simplified 88.3%

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

   1. associate-*r/ 88.1%

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

   2. *-commutative 88.1%

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

   3. associate-*r/ 88.1%

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

   4. associate-/r* 94.3%

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

   5. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

7. Taylor expanded in z around 0 53.7%

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

   1. neg-mul-1 53.7%

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

   2. distribute-rgt-neg-in 53.7%

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

   3. associate-*r/ 57.9%

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

   4. distribute-lft-out 61.4%

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

9. Simplified 61.4%

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

10. Taylor expanded in z around inf 27.2%

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

    1. mul-1-neg 27.2%

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

    2. distribute-frac-neg2 27.2%

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

    3. associate-/l* 32.4%

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

12. Simplified 32.4%

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

    1. clear-num 32.5%

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

    2. un-div-inv 32.5%

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

    3. add-sqr-sqrt 24.0%

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

    4. sqrt-unprod 53.5%

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

    5. sqr-neg 53.5%

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

    6. sqrt-unprod 16.4%

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

    7. add-sqr-sqrt 27.9%

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

14. Applied egg-rr 27.9%

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

15. Final simplification 27.9%

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

Alternative 13: 33.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-/l* 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-/r* 88.3%

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

   5. sqr-neg 88.3%

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

3. Simplified 88.3%

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

   1. associate-*r/ 88.1%

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

   2. *-commutative 88.1%

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

   3. associate-*r/ 88.1%

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

   4. associate-/r* 94.3%

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

   5. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

7. Taylor expanded in z around 0 53.7%

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

   1. neg-mul-1 53.7%

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

   2. distribute-rgt-neg-in 53.7%

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

   3. associate-*r/ 57.9%

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

   4. distribute-lft-out 61.4%

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

9. Simplified 61.4%

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

10. Taylor expanded in z around inf 27.2%

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

    1. mul-1-neg 27.2%

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

    2. distribute-frac-neg2 27.2%

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

    3. associate-/l* 32.4%

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

12. Simplified 32.4%

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

    1. associate-*r/ 27.2%

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

    2. add-sqr-sqrt 20.6%

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

    3. *-commutative 20.6%

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

    4. sqrt-unprod 52.0%

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

    5. sqr-neg 52.0%

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

    6. sqrt-unprod 14.2%

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

    7. add-sqr-sqrt 24.1%

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

    8. associate-*l/ 27.8%

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

    9. associate-/r/ 29.6%

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

14. Applied egg-rr 29.6%

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

15. Final simplification 29.6%

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

Developer target: 97.0% 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 2024039 
(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))))