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

Percentage Accurate: 83.7% → 97.5%

Time: 8.6s

Alternatives: 11

Speedup: 1.0×

• 25.3% 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: 83.7% 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: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

   1. *-commutative 86.9%

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

   2. associate-/l* 89.2%

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

   3. sqr-neg 89.2%

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

   4. associate-/r* 89.6%

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

   5. sqr-neg 89.6%

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

3. Simplified 89.6%

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

   1. associate-*r/ 91.1%

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

   2. *-commutative 91.1%

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

   3. associate-*r/ 91.4%

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

   4. associate-/r* 96.0%

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

   5. associate-*l/ 98.3%

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

6. Applied egg-rr 98.3%

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

Alternative 2: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x}{z + 1}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-125}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ x (+ z 1.0))))
   (*
    y_s
    (if (<= (/ (* x y_m) (* (+ z 1.0) (* z z))) 1e-125)
      (* (/ y_m (* z z)) t_0)
      (* (/ y_m z) (/ t_0 z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = x / (z + 1.0);
	double tmp;
	if (((x * y_m) / ((z + 1.0) * (z * z))) <= 1e-125) {
		tmp = (y_m / (z * z)) * t_0;
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (z + 1.0d0)
    if (((x * y_m) / ((z + 1.0d0) * (z * z))) <= 1d-125) then
        tmp = (y_m / (z * z)) * t_0
    else
        tmp = (y_m / z) * (t_0 / z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = x / (z + 1.0);
	double tmp;
	if (((x * y_m) / ((z + 1.0) * (z * z))) <= 1e-125) {
		tmp = (y_m / (z * z)) * t_0;
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	t_0 = x / (z + 1.0)
	tmp = 0
	if ((x * y_m) / ((z + 1.0) * (z * z))) <= 1e-125:
		tmp = (y_m / (z * z)) * t_0
	else:
		tmp = (y_m / z) * (t_0 / z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(x / Float64(z + 1.0))
	tmp = 0.0
	if (Float64(Float64(x * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 1e-125)
		tmp = Float64(Float64(y_m / Float64(z * z)) * t_0);
	else
		tmp = Float64(Float64(y_m / z) * Float64(t_0 / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = x / (z + 1.0);
	tmp = 0.0;
	if (((x * y_m) / ((z + 1.0) * (z * z))) <= 1e-125)
		tmp = (y_m / (z * z)) * t_0;
	else
		tmp = (y_m / z) * (t_0 / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(x * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-125], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.00000000000000001e-125

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. sqr-neg 92.4%

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

      3. times-frac 95.4%

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

      4. sqr-neg 95.4%

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

   3. Simplified 95.4%

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

   if 1.00000000000000001e-125 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. frac-times 79.0%

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

      3. associate-*l/ 80.8%

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

      4. times-frac 97.4%

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

   4. Applied egg-rr 97.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-125}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.00048\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x}}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.0) (not (<= z 0.00048)))
    (/ (* (/ x z) (/ y_m z)) z)
    (/ (/ y_m (/ z x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.00048)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x)) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.00048d0))) then
        tmp = ((x / z) * (y_m / z)) / z
    else
        tmp = (y_m / (z / x)) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.00048)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x)) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.00048):
		tmp = ((x / z) * (y_m / z)) / z
	else:
		tmp = (y_m / (z / x)) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.00048))
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(Float64(y_m / Float64(z / x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.00048)))
		tmp = ((x / z) * (y_m / z)) / z;
	else
		tmp = (y_m / (z / x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.00048]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 4.80000000000000012e-4 < z

   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* 89.6%

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

      3. sqr-neg 89.6%

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

      4. associate-/r* 90.3%

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

      5. sqr-neg 90.3%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r/ 93.8%

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

      4. associate-/r* 96.8%

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

      5. associate-*l/ 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around inf 97.4%

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

   if -1 < z < 4.80000000000000012e-4

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. times-frac 86.6%

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

      4. sqr-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around 0 85.8%

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

      1. associate-/r* 90.6%

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

      2. associate-*l/ 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 96.6%

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

      3. clear-num 96.6%

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

      4. un-div-inv 96.7%

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

   9. Applied egg-rr 96.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.00048\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.00048\right):\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x}}}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.0) (not (<= z 0.00048)))
    (* (/ y_m z) (/ (/ x z) z))
    (/ (/ y_m (/ z x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.00048)) {
		tmp = (y_m / z) * ((x / z) / z);
	} else {
		tmp = (y_m / (z / x)) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.00048d0))) then
        tmp = (y_m / z) * ((x / z) / z)
    else
        tmp = (y_m / (z / x)) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.00048)) {
		tmp = (y_m / z) * ((x / z) / z);
	} else {
		tmp = (y_m / (z / x)) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.00048):
		tmp = (y_m / z) * ((x / z) / z)
	else:
		tmp = (y_m / (z / x)) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.00048))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(y_m / Float64(z / x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.00048)))
		tmp = (y_m / z) * ((x / z) / z);
	else
		tmp = (y_m / (z / x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.00048]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 4.80000000000000012e-4 < z

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. frac-times 93.9%

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

      3. associate-*l/ 95.3%

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

      4. times-frac 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around inf 95.2%

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

   if -1 < z < 4.80000000000000012e-4

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. times-frac 86.6%

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

      4. sqr-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around 0 85.8%

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

      1. associate-/r* 90.6%

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

      2. associate-*l/ 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 96.6%

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

      3. clear-num 96.6%

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

      4. un-div-inv 96.7%

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

   9. Applied egg-rr 96.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.00048\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{elif}\;z \leq 0.00048:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{z}}{z}}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z -1.0)
    (/ (* (/ x z) (/ y_m z)) z)
    (if (<= z 0.00048) (/ (/ y_m (/ z x)) z) (/ (/ (/ y_m z) z) (/ z x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (z <= 0.00048) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = ((y_m / z) / z) / (z / x);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = ((x / z) * (y_m / z)) / z
    else if (z <= 0.00048d0) then
        tmp = (y_m / (z / x)) / z
    else
        tmp = ((y_m / z) / z) / (z / x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (z <= 0.00048) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = ((y_m / z) / z) / (z / x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= -1.0:
		tmp = ((x / z) * (y_m / z)) / z
	elif z <= 0.00048:
		tmp = (y_m / (z / x)) / z
	else:
		tmp = ((y_m / z) / z) / (z / x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	elseif (z <= 0.00048)
		tmp = Float64(Float64(y_m / Float64(z / x)) / z);
	else
		tmp = Float64(Float64(Float64(y_m / z) / z) / Float64(z / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = ((x / z) * (y_m / z)) / z;
	elseif (z <= 0.00048)
		tmp = (y_m / (z / x)) / z;
	else
		tmp = ((y_m / z) / z) / (z / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, -1.0], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.00048], N[(N[(y$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/l* 85.9%

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

      3. sqr-neg 85.9%

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

      4. associate-/r* 87.2%

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

      5. sqr-neg 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r/ 91.4%

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

      4. associate-/r* 95.8%

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

      5. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf 98.7%

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

   if -1 < z < 4.80000000000000012e-4

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. times-frac 86.6%

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

      4. sqr-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around 0 85.8%

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

      1. associate-/r* 90.6%

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

      2. associate-*l/ 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 96.6%

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

      3. clear-num 96.6%

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

      4. un-div-inv 96.7%

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

   9. Applied egg-rr 96.7%

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

   if 4.80000000000000012e-4 < z

   1. Initial program 93.8%

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

      1. *-commutative 93.8%

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

      2. frac-times 93.6%

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

      3. associate-*l/ 96.5%

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

      4. times-frac 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*r/ 98.1%

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

      2. frac-times 95.4%

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

      3. *-commutative 95.4%

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

      4. frac-times 98.1%

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

      5. associate-/l* 95.2%

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

      6. clear-num 95.0%

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

      7. associate-*l/ 95.1%

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

      8. *-un-lft-identity 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in z around inf 93.0%

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

Alternative 6: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.00048:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z -1.0)
    (* (/ y_m z) (/ (/ x z) z))
    (if (<= z 0.00048) (/ (/ y_m (/ z x)) z) (* (/ x z) (/ y_m (* z z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y_m / z) * ((x / z) / z);
	} else if (z <= 0.00048) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = (x / z) * (y_m / (z * z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (y_m / z) * ((x / z) / z)
    else if (z <= 0.00048d0) then
        tmp = (y_m / (z / x)) / z
    else
        tmp = (x / z) * (y_m / (z * z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y_m / z) * ((x / z) / z);
	} else if (z <= 0.00048) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = (x / z) * (y_m / (z * z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y_m / z) * ((x / z) / z)
	elif z <= 0.00048:
		tmp = (y_m / (z / x)) / z
	else:
		tmp = (x / z) * (y_m / (z * z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x / z) / z));
	elseif (z <= 0.00048)
		tmp = Float64(Float64(y_m / Float64(z / x)) / z);
	else
		tmp = Float64(Float64(x / z) * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y_m / z) * ((x / z) / z);
	elseif (z <= 0.00048)
		tmp = (y_m / (z / x)) / z;
	else
		tmp = (x / z) * (y_m / (z * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, -1.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00048], N[(N[(y$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. frac-times 94.1%

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

      3. associate-*l/ 94.1%

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

      4. times-frac 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in z around inf 94.6%

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

   if -1 < z < 4.80000000000000012e-4

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. times-frac 86.6%

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

      4. sqr-neg 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around 0 85.8%

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

      1. associate-/r* 90.6%

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

      2. associate-*l/ 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 96.6%

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

      3. clear-num 96.6%

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

      4. un-div-inv 96.7%

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

   9. Applied egg-rr 96.7%

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

   if 4.80000000000000012e-4 < z

   1. Initial program 93.8%

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

      1. *-commutative 93.8%

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

      2. sqr-neg 93.8%

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

      3. times-frac 93.6%

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

      4. sqr-neg 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in z around inf 91.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.00048:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2.5e-151) (/ (/ y_m (/ z x)) z) (* x (/ y_m (* z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.5e-151) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.5d-151) then
        tmp = (y_m / (z / x)) / z
    else
        tmp = x * (y_m / (z * z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.5e-151) {
		tmp = (y_m / (z / x)) / z;
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.5e-151:
		tmp = (y_m / (z / x)) / z
	else:
		tmp = x * (y_m / (z * z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.5e-151)
		tmp = Float64(Float64(y_m / Float64(z / x)) / z);
	else
		tmp = Float64(x * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.5e-151)
		tmp = (y_m / (z / x)) / z;
	else
		tmp = x * (y_m / (z * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.5e-151], N[(N[(y$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000002e-151

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. sqr-neg 86.0%

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

      3. times-frac 88.7%

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

      4. sqr-neg 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around 0 76.8%

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

      1. associate-/r* 78.2%

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

      2. associate-*l/ 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. associate-*l/ 75.0%

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

      2. associate-*r/ 82.4%

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

      3. clear-num 82.8%

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

      4. un-div-inv 82.8%

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

   9. Applied egg-rr 82.8%

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

   if 2.50000000000000002e-151 < x

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. sqr-neg 88.2%

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

      3. times-frac 93.0%

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

      4. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in z around 0 75.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 6e-152) (/ (* (/ x z) y_m) z) (* x (/ y_m (* z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 6e-152) {
		tmp = ((x / z) * y_m) / z;
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6d-152) then
        tmp = ((x / z) * y_m) / z
    else
        tmp = x * (y_m / (z * z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 6e-152) {
		tmp = ((x / z) * y_m) / z;
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 6e-152:
		tmp = ((x / z) * y_m) / z
	else:
		tmp = x * (y_m / (z * z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 6e-152)
		tmp = Float64(Float64(Float64(x / z) * y_m) / z);
	else
		tmp = Float64(x * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 6e-152)
		tmp = ((x / z) * y_m) / z;
	else
		tmp = x * (y_m / (z * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 6e-152], N[(N[(N[(x / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6e-152

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-/l* 87.4%

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

      3. sqr-neg 87.4%

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

      4. associate-/r* 88.0%

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

      5. sqr-neg 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r/ 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*r/ 89.8%

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

      4. associate-/r* 95.4%

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

      5. associate-*l/ 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in z around 0 82.4%

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

   if 6e-152 < x

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. sqr-neg 88.2%

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

      3. times-frac 93.0%

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

      4. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in z around 0 75.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2.5e-151) (* (/ x z) (/ y_m z)) (* x (/ y_m (* z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.5e-151) {
		tmp = (x / z) * (y_m / z);
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.5d-151) then
        tmp = (x / z) * (y_m / z)
    else
        tmp = x * (y_m / (z * z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.5e-151) {
		tmp = (x / z) * (y_m / z);
	} else {
		tmp = x * (y_m / (z * z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.5e-151:
		tmp = (x / z) * (y_m / z)
	else:
		tmp = x * (y_m / (z * z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.5e-151)
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	else
		tmp = Float64(x * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.5e-151)
		tmp = (x / z) * (y_m / z);
	else
		tmp = x * (y_m / (z * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.5e-151], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000002e-151

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. frac-times 88.7%

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

      3. associate-*l/ 89.1%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 82.4%

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

   if 2.50000000000000002e-151 < x

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. sqr-neg 88.2%

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

      3. times-frac 93.0%

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

      4. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in z around 0 75.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

   1. *-commutative 86.9%

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

   2. frac-times 90.4%

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

   3. associate-*l/ 91.0%

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

   4. times-frac 96.8%

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

4. Applied egg-rr 96.8%

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

Alternative 11: 74.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

   1. *-commutative 86.9%

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

   2. frac-times 90.4%

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

   3. associate-*l/ 91.0%

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

   4. times-frac 96.8%

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

4. Applied egg-rr 96.8%

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

5. Taylor expanded in z around 0 76.5%

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

6. Final simplification 76.5%

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

Developer target: 96.3% 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 2024102 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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