Average Error: 37.7 → 26.5
Time: 1.9s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.62541230777024819 \cdot 10^{124}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 8.8980375400702012 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le 2.64333370990503235 \cdot 10^{-167}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 2.2544939054547067 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original37.7
Target25.5
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.62541230777024819e124

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 17.5

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

    3. Simplified17.5

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

    if -1.62541230777024819e124 < x < 8.8980375400702012e-248 or 2.64333370990503235e-167 < x < 2.2544939054547067e84

    1. Initial program 28.6

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

    if 8.8980375400702012e-248 < x < 2.64333370990503235e-167

    1. Initial program 33.1

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

    2. Taylor expanded around 0 48.0

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

    if 2.2544939054547067e84 < x

    1. Initial program 53.3

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

    2. Taylor expanded around inf 19.2

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

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.62541230777024819 \cdot 10^{124}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 8.8980375400702012 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le 2.64333370990503235 \cdot 10^{-167}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 2.2544939054547067 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (neg z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

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