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

Average Error: 14.7 → 3.3

Time: 11.5s

Precision: binary64

Cost: 1736

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{z} \cdot y}{z + 1}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1}}{z \cdot \frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (if (<= t_0 -1e+19)
     (/ (* (/ (/ x z) z) y) (+ z 1.0))
     (if (<= t_0 2e-158)
       (/ (/ x (/ z y)) z)
       (/ (/ y (+ z 1.0)) (* z (/ z x)))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1e+19) {
		tmp = (((x / z) / z) * y) / (z + 1.0);
	} else if (t_0 <= 2e-158) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = (y / (z + 1.0)) / (z * (z / x));
	}
	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
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if (t_0 <= (-1d+19)) then
        tmp = (((x / z) / z) * y) / (z + 1.0d0)
    else if (t_0 <= 2d-158) then
        tmp = (x / (z / y)) / z
    else
        tmp = (y / (z + 1.0d0)) / (z * (z / x))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1e+19) {
		tmp = (((x / z) / z) * y) / (z + 1.0);
	} else if (t_0 <= 2e-158) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = (y / (z + 1.0)) / (z * (z / x));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if t_0 <= -1e+19:
		tmp = (((x / z) / z) * y) / (z + 1.0)
	elif t_0 <= 2e-158:
		tmp = (x / (z / y)) / z
	else:
		tmp = (y / (z + 1.0)) / (z * (z / x))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+19)
		tmp = Float64(Float64(Float64(Float64(x / z) / z) * y) / Float64(z + 1.0));
	elseif (t_0 <= 2e-158)
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	else
		tmp = Float64(Float64(y / Float64(z + 1.0)) / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+19)
		tmp = (((x / z) / z) * y) / (z + 1.0);
	elseif (t_0 <= 2e-158)
		tmp = (x / (z / y)) / z;
	else
		tmp = (y / (z + 1.0)) / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+19], N[(N[(N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-158], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	14.7
Target	4.0
Herbie	3.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -1e19

   1. Initial program 10.5

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

   2. Simplified 4.9

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

      [Start] 10.5	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.9	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.2

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

   if -1e19 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 2.00000000000000013e-158

   1. Initial program 34.9

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

   2. Simplified 19.9

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

      [Start] 34.9	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      *-commutative [=>] 34.9	\[ \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      associate-*r/ [<=] 36.6	\[ \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]
      associate-*l* [=>] 36.6	\[ y \cdot \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]
      associate-/r* [=>] 19.9	\[ y \cdot \color{blue}{\frac{\frac{x}{z}}{z \cdot \left(z + 1\right)}} \]
      distribute-rgt-in [=>] 19.9	\[ y \cdot \frac{\frac{x}{z}}{\color{blue}{z \cdot z + 1 \cdot z}} \]
      *-lft-identity [=>] 19.9	\[ y \cdot \frac{\frac{x}{z}}{z \cdot z + \color{blue}{z}} \]
      fma-def [=>] 19.9	\[ y \cdot \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   3. Taylor expanded in z around 0 36.9

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

   4. Simplified 8.4

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

      [Start] 36.9	\[ \frac{y \cdot x}{{z}^{2}} \]
      *-commutative [=>] 36.9	\[ \frac{\color{blue}{x \cdot y}}{{z}^{2}} \]
      unpow2 [=>] 36.9	\[ \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
      times-frac [=>] 8.4	\[ \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

   5. Applied egg-rr 7.0

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

   if 2.00000000000000013e-158 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 8.6

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

   2. Simplified 4.2

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

      [Start] 8.6	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.2	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.5

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{z} \cdot y}{z + 1}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-21} \lor \neg \left(z \leq 8 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{z} \cdot y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 2
Error	4.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 3
Error	6.1
Cost	841

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

Alternative 4
Error	4.4
Cost	841

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

Alternative 5
Error	4.5
Cost	840

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

Alternative 6
Error	4.4
Cost	840

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

Alternative 7
Error	4.2
Cost	840

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

Alternative 8
Error	17.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-94} \lor \neg \left(y \leq 3.1 \cdot 10^{+80}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9
Error	17.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+78}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 12
Error	17.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-102}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]

Alternative 13
Error	17.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]

Alternative 14
Error	16.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 15
Error	21.7
Cost	448

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

Alternative 16
Error	45.5
Cost	384

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

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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