Average Error: 31.0 → 17.6
Time: 1.5s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.3287772795912036 \cdot 10^{57}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 7.5384845989707597 \cdot 10^{108}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original31.0
Target17.3
Herbie17.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.3287772795912036e57

    1. Initial program 45.5

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

    2. Taylor expanded around -inf 12.3

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

    3. Simplified12.3

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

    if -2.3287772795912036e57 < x < 7.5384845989707597e108

    1. Initial program 21.0

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

    if 7.5384845989707597e108 < x

    1. Initial program 53.0

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.3287772795912036 \cdot 10^{57}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 7.5384845989707597 \cdot 10^{108}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))