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

Percentage Accurate: 82.8% → 99.0%

Time: 8.2s

Alternatives: 11

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.8% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x_m \cdot y_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{-x_m}{z \cdot \left(\frac{z}{y_m} \cdot \left(-1 - z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z + 1} \cdot \frac{x_m}{z}}{z}\\ \end{array}\right) \end{array} \]

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 5e-35)
     (/ (- x_m) (* z (* (/ z y_m) (- -1.0 z))))
     (/ (* (/ y_m (+ z 1.0)) (/ x_m z)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-35) {
		tmp = -x_m / (z * ((z / y_m) * (-1.0 - z)));
	} else {
		tmp = ((y_m / (z + 1.0)) * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * y_m) / ((z * z) * (z + 1.0d0))) <= 5d-35) then
        tmp = -x_m / (z * ((z / y_m) * ((-1.0d0) - z)))
    else
        tmp = ((y_m / (z + 1.0d0)) * (x_m / z)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-35) {
		tmp = -x_m / (z * ((z / y_m) * (-1.0 - z)));
	} else {
		tmp = ((y_m / (z + 1.0)) * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-35:
		tmp = -x_m / (z * ((z / y_m) * (-1.0 - z)))
	else:
		tmp = ((y_m / (z + 1.0)) * (x_m / z)) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e-35)
		tmp = Float64(Float64(-x_m) / Float64(z * Float64(Float64(z / y_m) * Float64(-1.0 - z))));
	else
		tmp = Float64(Float64(Float64(y_m / Float64(z + 1.0)) * 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 (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-35)
		tmp = -x_m / (z * ((z / y_m) * (-1.0 - z)));
	else
		tmp = ((y_m / (z + 1.0)) * (x_m / z)) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-35], N[((-x$95$m) / N[(z * N[(N[(z / y$95$m), $MachinePrecision] * N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 4.99999999999999964e-35

   1. Initial program 87.6%

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

      1. frac-times 91.1%

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

      2. associate-*l/ 88.8%

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

      3. times-frac 95.3%

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

   3. Applied egg-rr 95.3%

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

      1. *-commutative 95.3%

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

      2. clear-num 95.3%

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

      3. frac-2neg 95.3%

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

      4. frac-times 94.6%

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

      5. *-un-lft-identity 94.6%

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

      6. associate-/r/ 94.6%

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

   5. Applied egg-rr 94.6%

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

   if 4.99999999999999964e-35 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. times-frac 79.5%

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

      3. sqr-neg 79.5%

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

   3. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 93.8%

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

      3. associate-*r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{-x}{z \cdot \left(\frac{z}{y} \cdot \left(-1 - z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 2: 97.0% accurate, 0.8× speedup

Math

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

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 1.0)))
     (* (/ 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 <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		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 <= 1.0))
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	else
		tmp = Float64(Float64(y_m * 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 <= 1.0)))
		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, 1.0]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 86.6%

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

      1. frac-times 91.5%

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

      2. associate-*l/ 94.1%

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

      3. times-frac 96.3%

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

   3. Applied egg-rr 96.3%

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

   4. Taylor expanded in z around inf 94.9%

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

   if -1 < z < 1

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. times-frac 83.5%

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

      3. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 94.2%

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

      3. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*r/ 95.7%

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

   8. Simplified 95.7%

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

4. Final simplification 95.3%

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

Alternative 3: 96.6% accurate, 0.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (/ x_m (* z (* z (/ z y_m))))
     (if (<= z 1.0) (/ (* y_m (/ x_m z)) z) (* (/ 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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 1.0) {
		tmp = (y_m * (x_m / z)) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x_m / (z * (z * (z / y_m)))
    else if (z <= 1.0d0) then
        tmp = (y_m * (x_m / z)) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 1.0) {
		tmp = (y_m * (x_m / z)) / z;
	} 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):
	tmp = 0
	if z <= -1.0:
		tmp = x_m / (z * (z * (z / y_m)))
	elif z <= 1.0:
		tmp = (y_m * (x_m / z)) / z
	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)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x_m / Float64(z * Float64(z * Float64(z / y_m))));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / z);
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x_m / (z * (z * (z / y_m)));
	elseif (z <= 1.0)
		tmp = (y_m * (x_m / z)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(x$95$m / N[(z * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 87.2%

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

      1. frac-times 88.5%

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

      2. associate-*l/ 94.3%

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

      3. times-frac 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around inf 91.4%

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

      1. *-commutative 91.4%

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

      2. clear-num 91.4%

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

      3. frac-times 90.9%

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

      4. *-un-lft-identity 90.9%

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

      5. div-inv 90.9%

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

      6. clear-num 91.0%

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

   6. Applied egg-rr 91.0%

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

   if -1 < z < 1

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. times-frac 83.5%

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

      3. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 94.2%

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

      3. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*r/ 95.7%

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

   8. Simplified 95.7%

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

   if 1 < z

   1. Initial program 85.8%

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

      1. frac-times 95.9%

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

      2. associate-*l/ 93.9%

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

      3. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 99.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z}{y}\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 4: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x_m}{z \cdot \left(z \cdot \frac{z}{y_m}\right)}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y_m \cdot \left(\frac{x_m}{z} - x_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} \cdot \frac{\frac{y_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (/ x_m (* z (* z (/ z y_m))))
     (if (<= z 0.75)
       (/ (* y_m (- (/ x_m z) x_m)) z)
       (* (/ 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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 0.75) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x_m / (z * (z * (z / y_m)))
    else if (z <= 0.75d0) then
        tmp = (y_m * ((x_m / z) - x_m)) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 0.75) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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):
	tmp = 0
	if z <= -1.0:
		tmp = x_m / (z * (z * (z / y_m)))
	elif z <= 0.75:
		tmp = (y_m * ((x_m / z) - x_m)) / z
	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)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x_m / Float64(z * Float64(z * Float64(z / y_m))));
	elseif (z <= 0.75)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m / z) - x_m)) / z);
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x_m / (z * (z * (z / y_m)));
	elseif (z <= 0.75)
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(x$95$m / N[(z * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 87.2%

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

      1. frac-times 88.5%

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

      2. associate-*l/ 94.3%

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

      3. times-frac 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around inf 91.4%

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

      1. *-commutative 91.4%

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

      2. clear-num 91.4%

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

      3. frac-times 90.9%

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

      4. *-un-lft-identity 90.9%

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

      5. div-inv 90.9%

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

      6. clear-num 91.0%

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

   6. Applied egg-rr 91.0%

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

   if -1 < z < 0.75

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. times-frac 83.5%

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

      3. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 94.2%

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

      3. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*r/ 90.2%

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

      6. *-commutative 90.2%

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

      7. distribute-lft-out 96.8%

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

   8. Simplified 96.8%

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

   9. Taylor expanded in y around 0 96.8%

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

   if 0.75 < z

   1. Initial program 85.8%

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

      1. frac-times 95.9%

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

      2. associate-*l/ 93.9%

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

      3. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 99.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z}{y}\right)}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 5: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x_m}{z \cdot \left(z \cdot \frac{z}{y_m}\right)}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y_m \cdot \left(\frac{x_m}{z} - x_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{z} \cdot \frac{y_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (/ x_m (* z (* z (/ z y_m))))
     (if (<= z 0.76)
       (/ (* y_m (- (/ x_m z) x_m)) z)
       (/ (* (/ 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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 0.76) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x_m / (z * (z * (z / y_m)))
    else if (z <= 0.76d0) then
        tmp = (y_m * ((x_m / z) - x_m)) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = x_m / (z * (z * (z / y_m)));
	} else if (z <= 0.76) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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):
	tmp = 0
	if z <= -1.0:
		tmp = x_m / (z * (z * (z / y_m)))
	elif z <= 0.76:
		tmp = (y_m * ((x_m / z) - x_m)) / z
	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)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x_m / Float64(z * Float64(z * Float64(z / y_m))));
	elseif (z <= 0.76)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m / z) - x_m)) / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x_m / (z * (z * (z / y_m)));
	elseif (z <= 0.76)
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(x$95$m / N[(z * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 87.2%

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

      1. frac-times 88.5%

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

      2. associate-*l/ 94.3%

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

      3. times-frac 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around inf 91.4%

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

      1. *-commutative 91.4%

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

      2. clear-num 91.4%

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

      3. frac-times 90.9%

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

      4. *-un-lft-identity 90.9%

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

      5. div-inv 90.9%

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

      6. clear-num 91.0%

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

   6. Applied egg-rr 91.0%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. times-frac 83.5%

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

      3. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 94.2%

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

      3. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*r/ 90.2%

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

      6. *-commutative 90.2%

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

      7. distribute-lft-out 96.8%

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

   8. Simplified 96.8%

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

   9. Taylor expanded in y around 0 96.8%

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

   if 0.76000000000000001 < z

   1. Initial program 85.8%

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

      1. sqr-neg 85.8%

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

      2. times-frac 95.9%

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

      3. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z}{y}\right)}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 6: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x_m \cdot \frac{y_m}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y_m \cdot \left(\frac{x_m}{z} - x_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{z} \cdot \frac{y_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (/ (/ (* x_m (/ y_m z)) z) z)
     (if (<= z 0.76)
       (/ (* y_m (- (/ x_m z) x_m)) z)
       (/ (* (/ 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 tmp;
	if (z <= -1.0) {
		tmp = ((x_m * (y_m / z)) / z) / z;
	} else if (z <= 0.76) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = ((x_m * (y_m / z)) / z) / z
    else if (z <= 0.76d0) then
        tmp = (y_m * ((x_m / z) - x_m)) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = ((x_m * (y_m / z)) / z) / z;
	} else if (z <= 0.76) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} 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):
	tmp = 0
	if z <= -1.0:
		tmp = ((x_m * (y_m / z)) / z) / z
	elif z <= 0.76:
		tmp = (y_m * ((x_m / z) - x_m)) / z
	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)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(x_m * Float64(y_m / z)) / z) / z);
	elseif (z <= 0.76)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m / z) - x_m)) / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = ((x_m * (y_m / z)) / z) / z;
	elseif (z <= 0.76)
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(N[(N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 87.2%

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

      1. sqr-neg 87.2%

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

      2. times-frac 88.5%

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

      3. sqr-neg 88.5%

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

   3. Simplified 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-/r* 92.2%

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

      3. associate-*r/ 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. associate-*r/ 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in z around inf 96.0%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. times-frac 83.5%

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

      3. sqr-neg 83.5%

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

   3. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 94.2%

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

      3. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*r/ 90.2%

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

      6. *-commutative 90.2%

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

      7. distribute-lft-out 96.8%

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

   8. Simplified 96.8%

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

   9. Taylor expanded in y around 0 96.8%

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

   if 0.76000000000000001 < z

   1. Initial program 85.8%

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

      1. sqr-neg 85.8%

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

      2. times-frac 95.9%

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

      3. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 7: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. frac-times 87.2%

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

   2. associate-*l/ 85.1%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

4. Final simplification 96.0%

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

Alternative 8: 98.2% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (* (/ y_m (+ z 1.0)) (/ 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 / (z + 1.0)) * (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 / (z + 1.0d0)) * (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 / (z + 1.0)) * (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 / (z + 1.0)) * (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(Float64(Float64(y_m / Float64(z + 1.0)) * 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 / (z + 1.0)) * (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[(N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. sqr-neg 81.7%

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

   2. times-frac 87.2%

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

   3. sqr-neg 87.2%

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

3. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-/r* 94.8%

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

   3. associate-*r/ 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

Alternative 9: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.05 \cdot 10^{+58}:\\ \;\;\;\;\frac{x_m}{z} \cdot \frac{y_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 2.05e+58) (* (/ x_m 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 (y_m <= 2.05e+58) {
		tmp = (x_m / 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 (y_m <= 2.05d+58) then
        tmp = (x_m / 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 (y_m <= 2.05e+58) {
		tmp = (x_m / 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 y_m <= 2.05e+58:
		tmp = (x_m / 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 (y_m <= 2.05e+58)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.05e+58)
		tmp = (x_m / 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[LessEqual[y$95$m, 2.05e+58], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e58

   1. Initial program 80.7%

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

      1. frac-times 86.5%

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

      2. associate-*l/ 82.6%

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

      3. times-frac 96.8%

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

   3. Applied egg-rr 96.8%

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

   4. Taylor expanded in z around 0 79.7%

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

   if 2.05e58 < y

   1. Initial program 86.1%

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

      1. sqr-neg 86.1%

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

      2. times-frac 90.4%

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

      3. sqr-neg 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in z around 0 71.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10: 74.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. frac-times 87.2%

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

   2. associate-*l/ 85.1%

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

   3. times-frac 96.0%

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

3. Applied egg-rr 96.0%

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

4. Taylor expanded in z around 0 74.0%

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

5. Final simplification 74.0%

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

Alternative 11: 27.4% accurate, 1.8× speedup

Math

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

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 (* 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) {
	return y_s * (x_s * (x_m * (-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 * (-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 * (-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 * (-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(x_m * Float64(Float64(-y_m) / z))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (x_m * (-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[(x$95$m * N[((-y$95$m) / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. sqr-neg 81.7%

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

   2. times-frac 87.2%

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

   3. sqr-neg 87.2%

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

3. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-/r* 94.8%

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

   3. associate-*r/ 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in z around 0 55.5%

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

   1. mul-1-neg 55.5%

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

   2. distribute-lft-neg-out 55.5%

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

   3. +-commutative 55.5%

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

   4. *-commutative 55.5%

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

   5. associate-*r/ 61.3%

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

   6. *-commutative 61.3%

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

   7. distribute-lft-out 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in z around inf 22.8%

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

    1. associate-*r/ 22.8%

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

    2. mul-1-neg 22.8%

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

    3. *-commutative 22.8%

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

    4. distribute-lft-neg-in 22.8%

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

    5. associate-*l/ 28.4%

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

11. Simplified 28.4%

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

12. Final simplification 28.4%

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

Developer target: 97.1% accurate, 0.8× 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 2023340 
(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))))