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

Percentage Accurate: 83.5% → 97.8%

Time: 8.9s

Alternatives: 10

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.5% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z} \cdot \frac{y\_m}{z}}{z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m} \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ x_m z) (/ y_m z)) z)))
   (*
    x_s
    (*
     y_s
     (if (<= z -1.08e+46)
       t_0
       (if (<= z 2.4e+15) (/ y_m (* (/ z x_m) (fma z z z))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((x_m / z) * (y_m / z)) / z;
	double tmp;
	if (z <= -1.08e+46) {
		tmp = t_0;
	} else if (z <= 2.4e+15) {
		tmp = y_m / ((z / x_m) * fma(z, z, z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z)
	tmp = 0.0
	if (z <= -1.08e+46)
		tmp = t_0;
	elseif (z <= 2.4e+15)
		tmp = Float64(y_m / Float64(Float64(z / x_m) * fma(z, z, z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.08e+46], t$95$0, If[LessEqual[z, 2.4e+15], N[(y$95$m / N[(N[(z / x$95$m), $MachinePrecision] * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.07999999999999994e46 or 2.4e15 < z

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. unpow-prod-down N/A

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

      13. times-frac N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -1.07999999999999994e46 < z < 2.4e15

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. distribute-lft1-in N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{\frac{z}{x} \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{\left(\frac{z}{y\_m} \cdot z\right) \cdot z}\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+23}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ x_m (* (* (/ z y_m) z) z))) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -20000.0)
       t_0
       (if (<= t_1 5e-278)
         (* (/ y_m z) (/ x_m z))
         (if (<= t_1 1e+23) (* (/ x_m (* (fma z z z) z)) y_m) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m / (((z / y_m) * z) * z);
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-278) {
		tmp = (y_m / z) * (x_m / z);
	} else if (t_1 <= 1e+23) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m / Float64(Float64(Float64(z / y_m) * z) * z))
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-278)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (t_1 <= 1e+23)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m / N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 5e-278], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+23], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e4 or 9.9999999999999992e22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{{z}^{2}} \cdot z} \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{z}^{2} \cdot z}} \cdot x \]

      8. unpow2 N/A

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

      9. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 83.5%

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

   9. Applied rewrites 96.0%

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

   if -2e4 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999985e-278

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft1-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 4.99999999999999985e-278 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999999992e22

   1. Initial program 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -20000:\\ \;\;\;\;\frac{x}{\left(\frac{z}{y} \cdot z\right) \cdot z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 10^{+23}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{z}{y} \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.6× 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 2024230 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

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