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

Percentage Accurate: 83.1% → 97.6%

Time: 8.6s

Alternatives: 6

Speedup: 1.0×

• 25.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 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 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-/l* 86.6%

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

   3. sqr-neg 86.6%

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

   4. associate-/r* 89.9%

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

   5. sqr-neg 89.9%

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

3. Simplified 89.9%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. associate-*r/ 90.1%

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

   4. associate-/r* 95.0%

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

   5. associate-*l/ 99.4%

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

6. Applied egg-rr 99.4%

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

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \end{array} \end{array} \]

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 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (if (<= t_0 -1e+30)
      (/ (* (/ x z) (/ y_m z)) z)
      (if (<= t_0 5e-111)
        (/ (/ x z) (/ z y_m))
        (* (/ y_m (* z z)) (/ x (+ z 1.0))))))))

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 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -1e+30) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (t_0 <= 5e-111) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	}
	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 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-1d+30)) then
        tmp = ((x / z) * (y_m / z)) / z
    else if (t_0 <= 5d-111) then
        tmp = (x / z) / (z / y_m)
    else
        tmp = (y_m / (z * z)) * (x / (z + 1.0d0))
    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 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -1e+30) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (t_0 <= 5e-111) {
		tmp = (x / z) / (z / y_m);
	} else {
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	}
	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 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -1e+30:
		tmp = ((x / z) * (y_m / z)) / z
	elif t_0 <= 5e-111:
		tmp = (x / z) / (z / y_m)
	else:
		tmp = (y_m / (z * z)) * (x / (z + 1.0))
	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(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -1e+30)
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	elseif (t_0 <= 5e-111)
		tmp = Float64(Float64(x / z) / Float64(z / y_m));
	else
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	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 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -1e+30)
		tmp = ((x / z) * (y_m / z)) / z;
	elseif (t_0 <= 5e-111)
		tmp = (x / z) / (z / y_m);
	else
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	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[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e+30], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-111], N[(N[(x / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e30

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-/l* 88.8%

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

      3. sqr-neg 88.8%

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

      4. associate-/r* 93.9%

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

      5. sqr-neg 93.9%

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

   3. Simplified 93.9%

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

      1. associate-*r/ 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*r/ 94.5%

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

      4. associate-/r* 99.7%

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

      5. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf 99.5%

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

   if -1e30 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000003e-111

   1. Initial program 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-/l* 82.8%

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

      3. sqr-neg 82.8%

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

      4. associate-/r* 82.8%

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

      5. sqr-neg 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. associate-*r/ 82.8%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in z around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/l* 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. associate-*r/ 98.9%

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

       2. associate-*l/ 98.9%

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

       3. clear-num 98.9%

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

       4. associate-*l/ 99.1%

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

       5. *-un-lft-identity 99.1%

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

   11. Applied egg-rr 99.1%

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

   if 5.0000000000000003e-111 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. sqr-neg 85.0%

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

      3. times-frac 95.2%

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

      4. sqr-neg 95.2%

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

   3. Simplified 95.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y\_m}{z}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+51}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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) (/ y_m z)) z)))
   (*
    y_s
    (if (<= z -400000.0)
      t_0
      (if (<= z 9e-163)
        (/ (* (/ x z) y_m) z)
        (if (<= z 1.12e+51) (* y_m (/ (/ x (* z z)) (+ z 1.0))) t_0))))))

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) * (y_m / z)) / z;
	double tmp;
	if (z <= -400000.0) {
		tmp = t_0;
	} else if (z <= 9e-163) {
		tmp = ((x / z) * y_m) / z;
	} else if (z <= 1.12e+51) {
		tmp = y_m * ((x / (z * z)) / (z + 1.0));
	} else {
		tmp = t_0;
	}
	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) * (y_m / z)) / z
    if (z <= (-400000.0d0)) then
        tmp = t_0
    else if (z <= 9d-163) then
        tmp = ((x / z) * y_m) / z
    else if (z <= 1.12d+51) then
        tmp = y_m * ((x / (z * z)) / (z + 1.0d0))
    else
        tmp = t_0
    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) * (y_m / z)) / z;
	double tmp;
	if (z <= -400000.0) {
		tmp = t_0;
	} else if (z <= 9e-163) {
		tmp = ((x / z) * y_m) / z;
	} else if (z <= 1.12e+51) {
		tmp = y_m * ((x / (z * z)) / (z + 1.0));
	} else {
		tmp = t_0;
	}
	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) * (y_m / z)) / z
	tmp = 0
	if z <= -400000.0:
		tmp = t_0
	elif z <= 9e-163:
		tmp = ((x / z) * y_m) / z
	elif z <= 1.12e+51:
		tmp = y_m * ((x / (z * z)) / (z + 1.0))
	else:
		tmp = t_0
	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(Float64(Float64(x / z) * Float64(y_m / z)) / z)
	tmp = 0.0
	if (z <= -400000.0)
		tmp = t_0;
	elseif (z <= 9e-163)
		tmp = Float64(Float64(Float64(x / z) * y_m) / z);
	elseif (z <= 1.12e+51)
		tmp = Float64(y_m * Float64(Float64(x / Float64(z * z)) / Float64(z + 1.0)));
	else
		tmp = t_0;
	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) * (y_m / z)) / z;
	tmp = 0.0;
	if (z <= -400000.0)
		tmp = t_0;
	elseif (z <= 9e-163)
		tmp = ((x / z) * y_m) / z;
	elseif (z <= 1.12e+51)
		tmp = y_m * ((x / (z * z)) / (z + 1.0));
	else
		tmp = t_0;
	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[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, -400000.0], t$95$0, If[LessEqual[z, 9e-163], N[(N[(N[(x / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.12e+51], N[(y$95$m * N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -4e5 or 1.11999999999999992e51 < z

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 85.8%

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

      3. sqr-neg 85.8%

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

      4. associate-/r* 93.4%

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

      5. sqr-neg 93.4%

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

   3. Simplified 93.4%

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

      1. associate-*r/ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*r/ 93.8%

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

      4. associate-/r* 98.9%

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

      5. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf 99.7%

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

   if -4e5 < z < 8.9999999999999995e-163

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-/l* 82.4%

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

      3. sqr-neg 82.4%

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

      4. associate-/r* 82.4%

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

      5. sqr-neg 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-/r* 89.3%

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

      5. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0 98.7%

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

   if 8.9999999999999995e-163 < z < 1.11999999999999992e51

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-/l* 97.6%

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

      3. sqr-neg 97.6%

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

      4. associate-/r* 97.5%

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

      5. sqr-neg 97.5%

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

   3. Simplified 97.5%

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

Alternative 4: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -400000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y\_m}}\\ \end{array} \end{array} \]

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 -400000.0) (not (<= z 1.0)))
    (/ (* (/ x z) (/ y_m z)) z)
    (/ (/ x z) (/ z y_m)))))

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 <= -400000.0) || !(z <= 1.0)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (x / z) / (z / y_m);
	}
	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 <= (-400000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x / z) * (y_m / z)) / z
    else
        tmp = (x / z) / (z / y_m)
    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 <= -400000.0) || !(z <= 1.0)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (x / z) / (z / y_m);
	}
	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 <= -400000.0) or not (z <= 1.0):
		tmp = ((x / z) * (y_m / z)) / z
	else:
		tmp = (x / z) / (z / y_m)
	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 <= -400000.0) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(Float64(x / z) / Float64(z / y_m));
	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 <= -400000.0) || ~((z <= 1.0)))
		tmp = ((x / z) * (y_m / z)) / z;
	else
		tmp = (x / z) / (z / y_m);
	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, -400000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4e5 or 1 < z

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-/l* 87.0%

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

      3. sqr-neg 87.0%

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

      4. associate-/r* 93.9%

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

      5. sqr-neg 93.9%

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

   3. Simplified 93.9%

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

      1. associate-*r/ 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*r/ 94.2%

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

      4. associate-/r* 98.9%

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

      5. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around inf 97.8%

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

   if -4e5 < z < 1

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-/l* 86.2%

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

      3. sqr-neg 86.2%

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

      4. associate-/r* 86.2%

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

      5. sqr-neg 86.2%

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

   3. Simplified 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r/ 86.1%

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

      4. associate-/r* 91.3%

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

      5. associate-*l/ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in z around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-/l* 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. associate-*r/ 97.2%

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

       2. associate-*l/ 96.9%

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

       3. clear-num 96.9%

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

       4. associate-*l/ 97.3%

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

       5. *-un-lft-identity 97.3%

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

   11. Applied egg-rr 97.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -400000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -400000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y\_m}}\\ \end{array} \end{array} \]

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 -400000.0) (not (<= z 1.0)))
    (* (/ x z) (/ y_m (* z z)))
    (/ (/ x z) (/ z y_m)))))

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 <= -400000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = (x / z) / (z / y_m);
	}
	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 <= (-400000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * (y_m / (z * z))
    else
        tmp = (x / z) / (z / y_m)
    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 <= -400000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = (x / z) / (z / y_m);
	}
	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 <= -400000.0) or not (z <= 1.0):
		tmp = (x / z) * (y_m / (z * z))
	else:
		tmp = (x / z) / (z / y_m)
	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 <= -400000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) / Float64(z / y_m));
	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 <= -400000.0) || ~((z <= 1.0)))
		tmp = (x / z) * (y_m / (z * z));
	else
		tmp = (x / z) / (z / y_m);
	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, -400000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4e5 or 1 < z

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. sqr-neg 79.7%

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

      3. times-frac 94.7%

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

      4. sqr-neg 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in z around inf 92.8%

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

   if -4e5 < z < 1

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-/l* 86.2%

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

      3. sqr-neg 86.2%

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

      4. associate-/r* 86.2%

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

      5. sqr-neg 86.2%

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

   3. Simplified 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r/ 86.1%

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

      4. associate-/r* 91.3%

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

      5. associate-*l/ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in z around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-/l* 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. associate-*r/ 97.2%

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

       2. associate-*l/ 96.9%

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

       3. clear-num 96.9%

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

       4. associate-*l/ 97.3%

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

       5. *-un-lft-identity 97.3%

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

   11. Applied egg-rr 97.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -400000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 1.6× speedup

Math

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

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 (/ (/ x 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) {
	return y_s * (y_m * ((x / z) / 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 * ((x / z) / 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 * ((x / z) / 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 * ((x / z) / 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(y_m * Float64(Float64(x / z) / 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 * ((x / z) / 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[(y$95$m * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-/l* 86.6%

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

   3. sqr-neg 86.6%

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

   4. associate-/r* 89.9%

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

   5. sqr-neg 89.9%

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

3. Simplified 89.9%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. associate-*r/ 90.1%

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

   4. associate-/r* 95.0%

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

   5. associate-*l/ 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in z around 0 74.3%

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

   1. *-commutative 74.3%

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

   2. associate-/l* 75.2%

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

9. Applied egg-rr 75.2%

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

Developer target: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024087 
(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))))