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

Average Error: 37.6 → 0.4

Time: 5.8s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

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

Target

Original	37.6
Target	25.4
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 37.6

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

2. Applied egg-rr 0.9

   \[\leadsto \color{blue}{\mathsf{hypot}\left(z, \mathsf{hypot}\left(x, y\right)\right) \cdot \frac{1}{\sqrt{3}}} \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(z, \mathsf{hypot}\left(y, x\right)\right)}{\sqrt{3}}} \]
   Proof
   (/.f64 (hypot.f64 z (hypot.f64 y x)) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 y y) (*.f64 x x))))) (sqrt.f64 3)): 97 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) (*.f64 x x)))) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (sqrt.f64 (+.f64 (pow.f64 y 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (sqrt.f64 (Rewrite=> +-commutative_binary64 (+.f64 (pow.f64 x 2) (pow.f64 y 2))))) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (sqrt.f64 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) (pow.f64 y 2)))) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (sqrt.f64 (+.f64 (*.f64 x x) (Rewrite=> unpow2_binary64 (*.f64 y y))))) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (/.f64 (hypot.f64 z (Rewrite=> hypot-def_binary64 (hypot.f64 x y))) (sqrt.f64 3)): 0 points increase in error, 97 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (hypot.f64 z (hypot.f64 x y)) 1)) (sqrt.f64 3)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (hypot.f64 z (hypot.f64 x y)) (/.f64 1 (sqrt.f64 3)))): 105 points increase in error, 0 points decrease in error

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	17.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right) \cdot \sqrt{0.3333333333333333}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right) \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 2
Error	17.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(y, x\right)}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right) \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 3
Error	19.3
Cost	13188

\[\begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right) \cdot \sqrt{0.3333333333333333}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 4
Error	41.7
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 5
Error	45.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;z \leq 7.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 6
Error	52.1
Cost	6592

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

Reproduce

herbie shell --seed 2022326 
(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)))