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

Average Error: 15.0 → 2.9

Time: 10.0s

Precision: binary64

Cost: 7240

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{elif}\;z \leq 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z} \cdot x}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.1e-62)
   (/ (* (/ y z) (/ x (+ z 1.0))) z)
   (if (<= z 1e+70) (/ (/ y z) (/ (fma z z z) x)) (/ (/ (* (/ y z) x) z) 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 (z <= -6.1e-62) {
		tmp = ((y / z) * (x / (z + 1.0))) / z;
	} else if (z <= 1e+70) {
		tmp = (y / z) / (fma(z, z, z) / x);
	} else {
		tmp = (((y / z) * x) / z) / 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 (z <= -6.1e-62)
		tmp = Float64(Float64(Float64(y / z) * Float64(x / Float64(z + 1.0))) / z);
	elseif (z <= 1e+70)
		tmp = Float64(Float64(y / z) / Float64(fma(z, z, z) / x));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) * x) / z) / z);
	end
	return 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[z, -6.1e-62], N[(N[(N[(y / z), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1e+70], N[(N[(y / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	15.0
Target	4.0
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if z < -6.1e-62

   1. Initial program 9.6

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

   2. Simplified 4.6

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
      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)))): 48 points increase in error, 26 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 1.3

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

   if -6.1e-62 < z < 1.00000000000000007e70

   1. Initial program 22.8

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

   2. Simplified 5.7

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z (Rewrite<= *-lft-identity_binary64 (*.f64 1 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) (*.f64 1 z))))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 z 1))))): 0 points increase in error, 2 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 81 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 z (*.f64 z (+.f64 z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (+.f64 z 1)))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 5.5

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

   if 1.00000000000000007e70 < z

   1. Initial program 11.9

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

   2. Simplified 4.5

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z (Rewrite<= *-lft-identity_binary64 (*.f64 1 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) (*.f64 1 z))))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 z 1))))): 0 points increase in error, 2 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 81 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 z (*.f64 z (+.f64 z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (+.f64 z 1)))): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 4.5

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

   4. Simplified 4.5

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

   5. Applied egg-rr 1.6

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{elif}\;z \leq 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z} \cdot x}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.6
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \mathbf{if}\;y \cdot x \leq 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 0.0002:\\ \;\;\;\;\frac{\frac{\frac{y \cdot x}{z + 1}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.8
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{y}{z \cdot \frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 4
Error	6.0
Cost	840

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	6.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 0.78:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 6
Error	3.7
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	4.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z \cdot \frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 8
Error	3.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 9
Error	1.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \end{array} \]

Alternative 10
Error	18.1
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+122}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 12
Error	16.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-228}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 13
Error	18.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 14
Error	42.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 15
Error	22.6
Cost	448

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

Alternative 16
Error	45.8
Cost	320

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

Reproduce

herbie shell --seed 2022337 
(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))))