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

Average Error: 37.9 → 0.4

Time: 5.1s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (/ (hypot (hypot x 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(hypot(x, 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(Math.hypot(x, 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(math.hypot(x, 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(hypot(x, 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(hypot(x, 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[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] ^ 2 + z ^ 2], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.9
Target	25.8
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 37.9

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

2. Simplified 37.9

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

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022152 
(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)))