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

Average Error: 14.6 → 1.9

Time: 10.4s

Precision: binary64

Cost: 7240

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{elif}\;x \leq 2.1796060171718055 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z + 1\right) \cdot \frac{z}{y}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.044285376970324e-201)
   (/ (* (/ y z) (/ x (+ z 1.0))) z)
   (if (<= x 2.1796060171718055e-249)
     (/ y (/ z (/ x (fma z z z))))
     (/ (/ x (* (+ z 1.0) (/ 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 tmp;
	if (x <= -3.044285376970324e-201) {
		tmp = ((y / z) * (x / (z + 1.0))) / z;
	} else if (x <= 2.1796060171718055e-249) {
		tmp = y / (z / (x / fma(z, z, z)));
	} else {
		tmp = (x / ((z + 1.0) * (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)
	tmp = 0.0
	if (x <= -3.044285376970324e-201)
		tmp = Float64(Float64(Float64(y / z) * Float64(x / Float64(z + 1.0))) / z);
	elseif (x <= 2.1796060171718055e-249)
		tmp = Float64(y / Float64(z / Float64(x / fma(z, z, z))));
	else
		tmp = Float64(Float64(x / Float64(Float64(z + 1.0) * Float64(z / y))) / 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[x, -3.044285376970324e-201], N[(N[(N[(y / z), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 2.1796060171718055e-249], N[(y / N[(z / N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(z + 1.0), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	14.6
Target	3.9
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

2. if x < -3.04428537697032398e-201

   1. Initial program 14.7

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

   2. Applied egg-rr 9.6

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

   3. Applied egg-rr 1.4

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

   if -3.04428537697032398e-201 < x < 2.17960601717180546e-249

   1. Initial program 15.3

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

   2. Simplified 5.1

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      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))))): 0 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))))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 78 points increase in error, 22 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)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 4.1

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

   if 2.17960601717180546e-249 < x

   1. Initial program 13.7

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

   2. Applied egg-rr 10.7

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

   3. Applied egg-rr 1.9

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

   4. Applied egg-rr 1.7

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{elif}\;x \leq 2.1796060171718055 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z + 1\right) \cdot \frac{z}{y}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	968

\[\begin{array}{l} t_0 := \frac{\frac{x}{\left(z + 1\right) \cdot \frac{z}{y}}}{z}\\ \mathbf{if}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1796060171718055 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \mathbf{elif}\;x \leq 2.1796060171718055 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z + 1\right) \cdot \frac{z}{y}}}{z}\\ \end{array} \]

Alternative 3
Error	4.6
Cost	840

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

Alternative 4
Error	4.5
Cost	840

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

Alternative 5
Error	4.1
Cost	840

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

Alternative 6
Error	2.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq 8.368261588298714 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(z + 1\right) \cdot \frac{z}{y}}}{z}\\ \end{array} \]

Alternative 7
Error	17.9
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.556982192233024 \cdot 10^{+141}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	17.9
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.0625968839915564 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 10
Error	16.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;x \leq -3.044285376970324 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 11
Error	21.5
Cost	448

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

Reproduce

herbie shell --seed 2022300 
(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))))