Average Error: 31.6 → 17.9
Time: 2.7s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.3408035230771657 \cdot 10^{139}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -4.7866568598534521 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -2.79783213760014545 \cdot 10^{-306}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.28820419612964551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original31.6
Target17.8
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -9.3408035230771657e139

    1. Initial program 60.5

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around -inf 8.4

      \[\leadsto \color{blue}{-1 \cdot x}\]

    3. Simplified8.4

      \[\leadsto \color{blue}{-x}\]

    if -9.3408035230771657e139 < x < -4.7866568598534521e-266 or -2.79783213760014545e-306 < x < 1.28820419612964551e92

    1. Initial program 20.8

      \[\sqrt{x \cdot x + y \cdot y}\]

    if -4.7866568598534521e-266 < x < -2.79783213760014545e-306

    1. Initial program 31.2

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around 0 31.5

      \[\leadsto \color{blue}{y}\]

    if 1.28820419612964551e92 < x

    1. Initial program 50.9

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around inf 11.5

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.3408035230771657 \cdot 10^{139}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -4.7866568598534521 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -2.79783213760014545 \cdot 10^{-306}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.28820419612964551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (neg x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))