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

Average Error: 14.8 → 2.8

Time: 9.4s

Precision: binary64

Cost: 1737

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq 10^{-251} \lor \neg \left(t_0 \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \end{array} \]

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 (or (<= t_0 1e-251) (not (<= t_0 5e-60)))
     (/ (* (/ y (+ z 1.0)) (/ x z)) z)
     (/ x (/ z (/ y z))))))

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-251) || !(t_0 <= 5e-60)) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else {
		tmp = x / (z / (y / 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
    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-251) .or. (.not. (t_0 <= 5d-60))) then
        tmp = ((y / (z + 1.0d0)) * (x / z)) / z
    else
        tmp = x / (z / (y / z))
    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-251) || !(t_0 <= 5e-60)) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else {
		tmp = x / (z / (y / z));
	}
	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-251) or not (t_0 <= 5e-60):
		tmp = ((y / (z + 1.0)) * (x / z)) / z
	else:
		tmp = x / (z / (y / z))
	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-251) || !(t_0 <= 5e-60))
		tmp = Float64(Float64(Float64(y / Float64(z + 1.0)) * Float64(x / z)) / z);
	else
		tmp = Float64(x / Float64(z / Float64(y / z)));
	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-251) || ~((t_0 <= 5e-60)))
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	else
		tmp = x / (z / (y / z));
	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[Or[LessEqual[t$95$0, 1e-251], N[Not[LessEqual[t$95$0, 5e-60]], $MachinePrecision]], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.8
Target	4.0
Herbie	2.8

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.00000000000000002e-251 or 5.0000000000000001e-60 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 15.8

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

   2. Simplified 11.5

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      Proof
      (*.f64 (/.f64 x (*.f64 z z)) (/.f64 y (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))): 1 points increase in error, 1 points decrease in error

   3. Applied egg-rr 1.9

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

   if 1.00000000000000002e-251 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.0000000000000001e-60

   1. Initial program 9.3

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

   2. Simplified 7.8

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 x (*.f64 z z)) (/.f64 y (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))): 1 points increase in error, 1 points decrease in error

   3. Taylor expanded in z around 0 9.3

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

   4. Simplified 5.5

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]
      Proof
      (*.f64 (/.f64 x z) (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x y) (*.f64 z z))): 3 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) (*.f64 z z)): 0 points increase in error, 3 points decrease in error
      (/.f64 (*.f64 y x) (Rewrite<= unpow2_binary64 (pow.f64 z 2))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 7.2

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-251} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.8
Cost	2252

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{z \cdot \frac{z}{x}}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{y \cdot x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -490000000 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 3
Error	3.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -490000000 \lor \neg \left(z \leq 0.76\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 4
Error	5.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \end{array} \]

Alternative 5
Error	4.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;\frac{\frac{y}{z \cdot \frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 6
Error	17.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-125} \lor \neg \left(z \leq 5 \cdot 10^{-100}\right):\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7
Error	18.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 8
Error	19.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 9
Error	16.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 10
Error	22.3
Cost	448

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

Alternative 11
Error	45.8
Cost	384

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

Reproduce

herbie shell --seed 2022343 
(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))))