Average Error: 31.7 → 17.8
Time: 2.5s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 3.4364574278384972 \cdot 10^{-206}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original31.7
Target17.6
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -4.69108546213632265e113

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 10.1

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

    3. Simplified10.1

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

    if -4.69108546213632265e113 < x < -7.88936342124094937e-297 or 3.4364574278384972e-206 < x < 4.4095657239434897e116

    1. Initial program 19.6

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

    if -7.88936342124094937e-297 < x < 3.4364574278384972e-206

    1. Initial program 29.4

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

    2. Taylor expanded around 0 32.0

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

    if 4.4095657239434897e116 < x

    1. Initial program 55.0

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 3.4364574278384972 \cdot 10^{-206}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))