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

Average Error: 14.5 → 2.2

Time: 10.9s

Precision: binary64

Cost: 1224

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* (/ y z) (/ (/ x z) z))
   (if (<= (* x y) -5e-45)
     (/ (/ (* x y) (* z z)) (+ z 1.0))
     (/ (* (/ x z) (/ y (+ z 1.0))) 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 tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (y / z) * ((x / z) / z);
	} else if ((x * y) <= -5e-45) {
		tmp = ((x * y) / (z * z)) / (z + 1.0);
	} else {
		tmp = ((x / z) * (y / (z + 1.0))) / z;
	}
	return tmp;
}

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 tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = (y / z) * ((x / z) / z);
	} else if ((x * y) <= -5e-45) {
		tmp = ((x * y) / (z * z)) / (z + 1.0);
	} else {
		tmp = ((x / z) * (y / (z + 1.0))) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = (y / z) * ((x / z) / z)
	elif (x * y) <= -5e-45:
		tmp = ((x * y) / (z * z)) / (z + 1.0)
	else:
		tmp = ((x / z) * (y / (z + 1.0))) / 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)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	elseif (Float64(x * y) <= -5e-45)
		tmp = Float64(Float64(Float64(x * y) / Float64(z * z)) / Float64(z + 1.0));
	else
		tmp = Float64(Float64(Float64(x / z) * Float64(y / Float64(z + 1.0))) / 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)
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = (y / z) * ((x / z) / z);
	elseif ((x * y) <= -5e-45)
		tmp = ((x * y) / (z * z)) / (z + 1.0);
	else
		tmp = ((x / z) * (y / (z + 1.0))) / 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_] := If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-45], N[(N[(N[(x * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	14.5
Target	3.8
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 22.5

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      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)))): 55 points increase in error, 31 points decrease in error

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 1.1

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

   5. Taylor expanded in z around inf 1.2

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

   if -inf.0 < (*.f64 x y) < -4.99999999999999976e-45

   1. Initial program 6.5

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

   2. Simplified 7.0

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      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)))): 55 points increase in error, 31 points decrease in error

   3. Applied egg-rr 5.1

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

   4. Taylor expanded in x around 0 6.5

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2} \cdot \left(1 + z\right)}} \]

   5. Simplified 2.0

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{z \cdot z}}{z + 1}} \]
      Proof
      (/.f64 (/.f64 (*.f64 y x) (*.f64 z z)) (+.f64 z 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 y x) (Rewrite<= unpow2_binary64 (pow.f64 z 2))) (+.f64 z 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 y x) (pow.f64 z 2)) (Rewrite<= +-commutative_binary64 (+.f64 1 z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 y x) (*.f64 (pow.f64 z 2) (+.f64 1 z)))): 18 points increase in error, 8 points decrease in error

   if -4.99999999999999976e-45 < (*.f64 x y)

   1. Initial program 14.5

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

   2. Simplified 11.4

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      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)))): 55 points increase in error, 31 points decrease in error

   3. Applied egg-rr 2.3

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	2248

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

Alternative 2
Error	4.6
Cost	1744

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 3
Error	3.6
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \end{array} \]

Alternative 4
Error	3.0
Cost	968

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	6.1
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 6
Error	4.2
Cost	840

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

Alternative 7
Error	4.0
Cost	840

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

Alternative 8
Error	4.3
Cost	840

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

Alternative 9
Error	18.0
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-159}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-185}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 12
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-187}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 13
Error	18.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 14
Error	18.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 15
Error	43.4
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16
Error	42.5
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{z}{-y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 17
Error	23.6
Cost	448

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

Alternative 18
Error	46.0
Cost	384

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

Reproduce

herbie shell --seed 2022330 
(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))))