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

Average Error: 14.6 → 3.1

Time: 8.6s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x (+ z 1.0)) (/ (/ y z) z))))
   (if (<= z -1.4e-72) t_0 (if (<= z 3e-113) (/ (/ x (/ z y)) z) t_0))))

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 = (x / (z + 1.0)) * ((y / z) / z);
	double tmp;
	if (z <= -1.4e-72) {
		tmp = t_0;
	} else if (z <= 3e-113) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = t_0;
	}
	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 = (x / (z + 1.0d0)) * ((y / z) / z)
    if (z <= (-1.4d-72)) then
        tmp = t_0
    else if (z <= 3d-113) then
        tmp = (x / (z / y)) / z
    else
        tmp = t_0
    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 = (x / (z + 1.0)) * ((y / z) / z);
	double tmp;
	if (z <= -1.4e-72) {
		tmp = t_0;
	} else if (z <= 3e-113) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x / (z + 1.0)) * ((y / z) / z)
	tmp = 0
	if z <= -1.4e-72:
		tmp = t_0
	elif z <= 3e-113:
		tmp = (x / (z / y)) / z
	else:
		tmp = t_0
	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(x / Float64(z + 1.0)) * Float64(Float64(y / z) / z))
	tmp = 0.0
	if (z <= -1.4e-72)
		tmp = t_0;
	elseif (z <= 3e-113)
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	else
		tmp = t_0;
	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 = (x / (z + 1.0)) * ((y / z) / z);
	tmp = 0.0;
	if (z <= -1.4e-72)
		tmp = t_0;
	elseif (z <= 3e-113)
		tmp = (x / (z / y)) / z;
	else
		tmp = t_0;
	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[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.4e-72], t$95$0, If[LessEqual[z, 3e-113], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Target

Original	14.6
Target	3.9
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e-72 or 3.0000000000000001e-113 < z

   1. Initial program 9.3

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

   2. Simplified 4.5

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
      Proof
      (*.f64 (/.f64 y (*.f64 z z)) (/.f64 x (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 (*.f64 z z) (+.f64 z 1)))): 52 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 (*.f64 z z) (+.f64 z 1))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.7

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

   4. Applied egg-rr 2.5

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

   if -1.3999999999999999e-72 < z < 3.0000000000000001e-113

   1. Initial program 38.6

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

   2. Simplified 40.3

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
      Proof
      (*.f64 (/.f64 y (*.f64 z z)) (/.f64 x (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 (*.f64 z z) (+.f64 z 1)))): 52 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 (*.f64 z z) (+.f64 z 1))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 6.5

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

   4. Taylor expanded in z around 0 6.5

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

   5. Applied egg-rr 6.0

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	968

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	5.8
Cost	840

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	5.9
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.2
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	3.8
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.6
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 8
Error	17.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 9
Error	2.6
Cost	704

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

Alternative 10
Error	17.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 11
Error	43.8
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	22.5
Cost	448

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

Alternative 13
Error	45.9
Cost	384

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

Reproduce

herbie shell --seed 2022325 
(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))))