Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1

Average Error: 37.4 → 0.4

Time: 9.8s

Precision: binary64

Cost: 19520

Math

\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]

↓

\[\frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)}{\sqrt{3}} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))

↓

(FPCore (x y z) :precision binary64 (/ (hypot x (hypot y z)) (sqrt 3.0)))

C

double code(double x, double y, double z) {
	return sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

↓

double code(double x, double y, double z) {
	return hypot(x, hypot(y, z)) / sqrt(3.0);
}

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

↓

public static double code(double x, double y, double z) {
	return Math.hypot(x, Math.hypot(y, z)) / Math.sqrt(3.0);
}

Python

def code(x, y, z):
	return math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))

↓

def code(x, y, z):
	return math.hypot(x, math.hypot(y, z)) / math.sqrt(3.0)

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0))
end

↓

function code(x, y, z)
	return Float64(hypot(x, hypot(y, z)) / sqrt(3.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
end

↓

function tmp = code(x, y, z)
	tmp = hypot(x, hypot(y, z)) / sqrt(3.0);
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[Sqrt[x ^ 2 + N[Sqrt[y ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Target

Original	37.4
Target	25.1
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \]

Derivation

1. Initial program 37.4

   \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]

2. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)}{\sqrt{3}}} \]

3. Final simplification 0.4

   \[\leadsto \frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)}{\sqrt{3}} \]

Alternatives

Alternative 1
Error	33.0
Cost	14876

\[\begin{array}{l} t_0 := \frac{z}{\sqrt{3}}\\ t_1 := z \cdot \sqrt{0.3333333333333333}\\ t_2 := \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 10^{+298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(y, x\right)}{\sqrt{3}}\\ \end{array} \]

Alternative 2
Error	20.2
Cost	13056

\[\frac{\mathsf{hypot}\left(z, x\right)}{\sqrt{3}} \]

Alternative 3
Error	20.2
Cost	13056

\[\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(z, x\right) \]

Alternative 4
Error	40.9
Cost	8924

\[\begin{array}{l} t_0 := \frac{z}{\sqrt{3}}\\ t_1 := z \cdot \sqrt{0.3333333333333333}\\ t_2 := \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 10^{+298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-0.5 \cdot \frac{y}{\frac{x}{y}} - x\right)\\ \end{array} \]

Alternative 5
Error	44.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0928096931712628 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-0.5 \cdot \frac{y}{\frac{x}{y}} - x\right)\\ \mathbf{elif}\;x \leq -5.349477531016157 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq -2.994381464941496 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{x \cdot x}{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 6
Error	44.2
Cost	7116

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0928096931712628 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5.349477531016157 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq -2.994381464941496 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{x \cdot x}{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 7
Error	44.3
Cost	7052

\[\begin{array}{l} t_0 := \frac{-x}{\sqrt{3}}\\ \mathbf{if}\;x \leq -1.0928096931712628 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.349477531016157 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq -119233.22702540075:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 8
Error	44.2
Cost	7052

\[\begin{array}{l} t_0 := \sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.0928096931712628 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.349477531016157 \cdot 10^{+32}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq -2.994381464941496 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 9
Error	51.7
Cost	6592

\[\frac{z}{\sqrt{3}} \]

Alternative 10
Error	51.7
Cost	6592

\[z \cdot \sqrt{0.3333333333333333} \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))