Average Error: 37.4 → 18.3

Time: 27.0s

Precision: 64

Internal precision: 128

\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3.0}}\]

⬇

\[\begin{array}{l} \mathbf{if}\;y \le -1.2461039277906793 \cdot 10^{+90}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{0.3333333333333333}\\ \mathbf{if}\;y \le -3833755296400741.5:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot x\\ \mathbf{if}\;y \le 2.7533805383758474 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{{y}^2 + \left({z}^2 + x \cdot x\right)} \cdot \sqrt{\frac{1}{3.0}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot y\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	37.4
Comparison	29.2
Herbie	18.3

\[ \begin{array}{l} \mathbf{if}\;z \lt -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3.0}}\\ \mathbf{if}\;z \lt 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left({z}^2 + {x}^2\right) + {y}^2}}{\sqrt{3.0}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \]

Derivation

Split input into 4 regimes.
if y < -1.2461039277906793e+90
1. Initial program 51.5
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3.0}}\]
2. Applied simplify 51.5
  \[\leadsto \color{blue}{\sqrt{\frac{{y}^2 + \left({z}^2 + x \cdot x\right)}{3.0}}}\]
3. Applied taylor 0.2
  \[\leadsto -1 \cdot \left(\sqrt{0.3333333333333333} \cdot y\right)\]
4. Taylor expanded around -inf 0.2
  
  \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{0.3333333333333333} \cdot y\right)}\]
5. Applied simplify 0.2
  \[\leadsto \color{blue}{\left(-y\right) \cdot \sqrt{0.3333333333333333}}\]
if -1.2461039277906793e+90 < y < -3833755296400741.5
1. Initial program 58.2
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3.0}}\]
2. Applied simplify 58.2
  \[\leadsto \color{blue}{\sqrt{\frac{{y}^2 + \left({z}^2 + x \cdot x\right)}{3.0}}}\]
3. Applied taylor 0.2
  \[\leadsto \sqrt{0.3333333333333333} \cdot x\]
4. Taylor expanded around 0 0.2
  
  \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot x}\]
if -3833755296400741.5 < y < 2.7533805383758474e+165
1. Initial program 27.8
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3.0}}\]
2. Applied simplify 27.8
  \[\leadsto \color{blue}{\sqrt{\frac{{y}^2 + \left({z}^2 + x \cdot x\right)}{3.0}}}\]
3. Using strategy rm
4. Applied div-inv 27.8
  \[\leadsto \sqrt{\color{blue}{\left({y}^2 + \left({z}^2 + x \cdot x\right)\right) \cdot \frac{1}{3.0}}}\]
5. Applied sqrt-prod 27.9
  \[\leadsto \color{blue}{\sqrt{{y}^2 + \left({z}^2 + x \cdot x\right)} \cdot \sqrt{\frac{1}{3.0}}}\]
if 2.7533805383758474e+165 < y
1. Initial program 59.6
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3.0}}\]
2. Applied simplify 59.6
  \[\leadsto \color{blue}{\sqrt{\frac{{y}^2 + \left({z}^2 + x \cdot x\right)}{3.0}}}\]
3. Applied taylor 0.2
  \[\leadsto \sqrt{0.3333333333333333} \cdot y\]
4. Taylor expanded around inf 0.2
  
  \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot y}\]
Recombined 4 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(851321798 3617685590 4011722190 1634036316 762064292 2712077172)'
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"

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

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

Error

Target

Derivation

`if y < -1.2461039277906793e+90`

`if -1.2461039277906793e+90 < y < -3833755296400741.5`

`if -3833755296400741.5 < y < 2.7533805383758474e+165`

`if 2.7533805383758474e+165 < y`

Runtime