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

Average Accuracy: 40.1% → 99.4%

Time: 9.3s

Precision: binary64

Cost: 38976

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)\\ \sqrt{t_0} \cdot \sqrt{t_0 \cdot 0.3333333333333333} \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot x (hypot z y))))
   (* (sqrt t_0) (sqrt (* t_0 0.3333333333333333)))))

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) {
	double t_0 = hypot(x, hypot(z, y));
	return sqrt(t_0) * sqrt((t_0 * 0.3333333333333333));
}

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) {
	double t_0 = Math.hypot(x, Math.hypot(z, y));
	return Math.sqrt(t_0) * Math.sqrt((t_0 * 0.3333333333333333));
}

Python

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

↓

def code(x, y, z):
	t_0 = math.hypot(x, math.hypot(z, y))
	return math.sqrt(t_0) * math.sqrt((t_0 * 0.3333333333333333))

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)
	t_0 = hypot(x, hypot(z, y))
	return Float64(sqrt(t_0) * sqrt(Float64(t_0 * 0.3333333333333333)))
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)
	t_0 = hypot(x, hypot(z, y));
	tmp = sqrt(t_0) * sqrt((t_0 * 0.3333333333333333));
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_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	40.1%
Target	59.4%
Herbie	99.4%

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

Derivation

1. Initial program 40.1%

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

2. Applied egg-rr 98.7%

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

   [Start] 40.1	\[ \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
   sqrt-div [=>] 40.0	\[ \color{blue}{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}} \]
   div-inv [=>] 39.7	\[ \color{blue}{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z} \cdot \frac{1}{\sqrt{3}}} \]
   associate-+l+ [=>] 39.7	\[ \sqrt{\color{blue}{x \cdot x + \left(y \cdot y + z \cdot z\right)}} \cdot \frac{1}{\sqrt{3}} \]
   add-sqr-sqrt [=>] 39.7	\[ \sqrt{x \cdot x + \color{blue}{\sqrt{y \cdot y + z \cdot z} \cdot \sqrt{y \cdot y + z \cdot z}}} \cdot \frac{1}{\sqrt{3}} \]
   hypot-def [=>] 54.1	\[ \color{blue}{\mathsf{hypot}\left(x, \sqrt{y \cdot y + z \cdot z}\right)} \cdot \frac{1}{\sqrt{3}} \]
   hypot-def [=>] 98.7	\[ \mathsf{hypot}\left(x, \color{blue}{\mathsf{hypot}\left(y, z\right)}\right) \cdot \frac{1}{\sqrt{3}} \]

3. Simplified 99.4%

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

   [Start] 98.7	\[ \mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \cdot \frac{1}{\sqrt{3}} \]
   associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \cdot 1}{\sqrt{3}}} \]
   *-rgt-identity [=>] 99.4	\[ \frac{\color{blue}{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)}}{\sqrt{3}} \]
   hypot-def [<=] 54.5	\[ \frac{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot y + z \cdot z}}\right)}{\sqrt{3}} \]
   unpow2 [<=] 54.5	\[ \frac{\mathsf{hypot}\left(x, \sqrt{\color{blue}{{y}^{2}} + z \cdot z}\right)}{\sqrt{3}} \]
   unpow2 [<=] 54.5	\[ \frac{\mathsf{hypot}\left(x, \sqrt{{y}^{2} + \color{blue}{{z}^{2}}}\right)}{\sqrt{3}} \]
   +-commutative [<=] 54.5	\[ \frac{\mathsf{hypot}\left(x, \sqrt{\color{blue}{{z}^{2} + {y}^{2}}}\right)}{\sqrt{3}} \]
   unpow2 [=>] 54.5	\[ \frac{\mathsf{hypot}\left(x, \sqrt{\color{blue}{z \cdot z} + {y}^{2}}\right)}{\sqrt{3}} \]
   unpow2 [=>] 54.5	\[ \frac{\mathsf{hypot}\left(x, \sqrt{z \cdot z + \color{blue}{y \cdot y}}\right)}{\sqrt{3}} \]
   hypot-def [=>] 99.4	\[ \frac{\mathsf{hypot}\left(x, \color{blue}{\mathsf{hypot}\left(z, y\right)}\right)}{\sqrt{3}} \]

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}\right)} \]
   Proof

   [Start] 99.4	\[ \frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)}{\sqrt{3}} \]
   div-inv [=>] 98.7	\[ \color{blue}{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right) \cdot \frac{1}{\sqrt{3}}} \]
   add-sqr-sqrt [=>] 98.7	\[ \color{blue}{\left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)}\right)} \cdot \frac{1}{\sqrt{3}} \]
   associate-*l* [=>] 98.7	\[ \color{blue}{\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \frac{1}{\sqrt{3}}\right)} \]
   pow1/2 [=>] 98.7	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \frac{1}{\color{blue}{{3}^{0.5}}}\right) \]
   pow-flip [=>] 99.1	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \color{blue}{{3}^{\left(-0.5\right)}}\right) \]
   metadata-eval [=>] 99.1	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{\color{blue}{-0.5}}\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right) \cdot 0.3333333333333333}} \]
   Proof

   [Start] 99.1	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}\right) \]
   add-sqr-sqrt [=>] 98.8	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}} \cdot \sqrt{\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}}\right)} \]
   sqrt-unprod [=>] 99.1	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \color{blue}{\sqrt{\left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot {3}^{-0.5}\right)}} \]
   swap-sqr [=>] 99.1	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)}\right) \cdot \left({3}^{-0.5} \cdot {3}^{-0.5}\right)}} \]
   add-sqr-sqrt [<=] 99.4	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\color{blue}{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \left({3}^{-0.5} \cdot {3}^{-0.5}\right)} \]
   pow-prod-up [=>] 99.4	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right) \cdot \color{blue}{{3}^{\left(-0.5 + -0.5\right)}}} \]
   metadata-eval [=>] 99.4	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right) \cdot {3}^{\color{blue}{-1}}} \]
   metadata-eval [=>] 99.4	\[ \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right) \cdot \color{blue}{0.3333333333333333}} \]

6. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	19520

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

Alternative 2
Accuracy	44.8%
Cost	14356

\[\begin{array}{l} t_0 := z \cdot \sqrt{0.3333333333333333}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 50000000000:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot 0.3333333333333333\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{0.3333333333333333 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \end{array} \]

Alternative 3
Accuracy	67.0%
Cost	13056

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

Alternative 4
Accuracy	28.3%
Cost	7956

\[\begin{array}{l} t_0 := z \cdot \sqrt{0.3333333333333333}\\ t_1 := \sqrt{0.3333333333333333 \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 50000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \end{array} \]

Alternative 5
Accuracy	28.3%
Cost	7956

\[\begin{array}{l} t_0 := z \cdot \sqrt{0.3333333333333333}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 50000000000:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot 0.3333333333333333\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{0.3333333333333333 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \end{array} \]

Alternative 6
Accuracy	29.4%
Cost	7053

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+57} \lor \neg \left(x \leq -12500000\right) \land x \leq -4 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 7
Accuracy	29.5%
Cost	7053

\[\begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -210000 \lor \neg \left(x \leq -5.2 \cdot 10^{-30}\right):\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \end{array} \]

Alternative 8
Accuracy	18.3%
Cost	6592

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

Reproduce

herbie shell --seed 2023147 
(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)))