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

Average Error: 15.3 → 2.5

Time: 17.5s

Precision: binary64

Cost: 8016

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ (* x y) (fma z z z)) z)))
   (if (<= (* x y) -1e+267)
     (/ (/ (/ y (/ z x)) z) z)
     (if (<= (* x y) -2e-264)
       t_0
       (if (<= (* x y) 4e-222)
         (/ (/ x (fma z z z)) (/ z y))
         (if (<= (* x y) 4e+186) t_0 (* (/ y (* z z)) (/ x 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 = ((x * y) / fma(z, z, z)) / z;
	double tmp;
	if ((x * y) <= -1e+267) {
		tmp = ((y / (z / x)) / z) / z;
	} else if ((x * y) <= -2e-264) {
		tmp = t_0;
	} else if ((x * y) <= 4e-222) {
		tmp = (x / fma(z, z, z)) / (z / y);
	} else if ((x * y) <= 4e+186) {
		tmp = t_0;
	} else {
		tmp = (y / (z * z)) * (x / 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(Float64(x * y) / fma(z, z, z)) / z)
	tmp = 0.0
	if (Float64(x * y) <= -1e+267)
		tmp = Float64(Float64(Float64(y / Float64(z / x)) / z) / z);
	elseif (Float64(x * y) <= -2e-264)
		tmp = t_0;
	elseif (Float64(x * y) <= 4e-222)
		tmp = Float64(Float64(x / fma(z, z, z)) / Float64(z / y));
	elseif (Float64(x * y) <= 4e+186)
		tmp = t_0;
	else
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / 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_] := Block[{t$95$0 = N[(N[(N[(x * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+267], N[(N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-264], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 4e-222], N[(N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+186], t$95$0, N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	15.3
Target	4.1
Herbie	2.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -9.9999999999999997e266

   1. Initial program 55.0

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

   2. Taylor expanded in z around inf 55.5

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{3}}} \]

   3. Simplified 29.5

      \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]
      Proof
      (*.f64 x (/.f64 y (pow.f64 z 3))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) (pow.f64 z 3))): 24 points increase in error, 30 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) (pow.f64 z 3)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 52.8

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

   5. Applied egg-rr 16.6

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

   6. Applied egg-rr 14.1

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

   if -9.9999999999999997e266 < (*.f64 x y) < -2e-264 or 4.00000000000000019e-222 < (*.f64 x y) < 3.99999999999999992e186

   1. Initial program 6.4

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      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))))): 3 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 56 points increase in error, 28 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, 1 points decrease in error

   3. Applied egg-rr 0.5

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

   if -2e-264 < (*.f64 x y) < 4.00000000000000019e-222

   1. Initial program 21.9

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      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))))): 3 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 56 points increase in error, 28 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, 1 points decrease in error

   3. Applied egg-rr 0.2

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

   if 3.99999999999999992e186 < (*.f64 x y)

   1. Initial program 38.1

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

   2. Taylor expanded in z around inf 41.8

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{3}}} \]

   3. Simplified 27.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]
      Proof
      (*.f64 x (/.f64 y (pow.f64 z 3))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) (pow.f64 z 3))): 24 points increase in error, 30 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) (pow.f64 z 3)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 36.9

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

   5. Applied egg-rr 17.4

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.2
Cost	8520

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ t_1 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.5
Cost	8016

\[\begin{array}{l} t_0 := \frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	5.2
Cost	7104

\[\frac{1}{\frac{\mathsf{fma}\left(z, z, z\right)}{x \cdot \frac{y}{z}}} \]

Alternative 4
Error	4.5
Cost	1736

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;t_0 \leq 10^{-75}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{1 + z}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -72964.68362894311:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;z \leq 10^{-56}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 6
Error	4.9
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{if}\;z \leq -72964.68362894311:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00011851717776881669:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	4.7
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{if}\;z \leq -72964.68362894311:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00011851717776881669:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	4.9
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{if}\;z \leq -72964.68362894311:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00011851717776881669:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	18.0
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	18.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 5.057726298388368 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 11
Error	18.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.5907947935764446 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	17.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3315941908409583 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 13
Error	17.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3315941908409583 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	43.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1066616644197048 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 15
Error	43.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1066616644197048 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 16
Error	22.2
Cost	448

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

Alternative 17
Error	49.2
Cost	320

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

Alternative 18
Error	46.5
Cost	320

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

Reproduce

herbie shell --seed 2022318 
(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))))