Average Error: 31.2 → 17.9
Time: 18.1s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.28654588335027985 \cdot 10^{127}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -1.34702897925383054 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 4.32762173516280031 \cdot 10^{-186}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original31.2
Target17.1
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 < -1.28654588335027985e127

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -1.28654588335027985e127 < x < -1.34702897925383054e-257 or 4.32762173516280031e-186 < x < 3.46617861303372194e36

    1. Initial program 18.5

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

    if -1.34702897925383054e-257 < x < 4.32762173516280031e-186

    1. Initial program 29.9

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

    2. Taylor expanded around 0 33.3

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

    if 3.46617861303372194e36 < x

    1. Initial program 43.9

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

    2. Taylor expanded around inf 12.7

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.28654588335027985 \cdot 10^{127}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -1.34702897925383054 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 4.32762173516280031 \cdot 10^{-186}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))