Average Error: 21.2 → 0.5
Time: 1.5s
Precision: binary64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.9999582525827406 \cdot 10^{135}:\\ \;\;\;\;\left(-x\right) + y \cdot \frac{\frac{-1}{2}}{x}\\ \mathbf{elif}\;x \le 5.73985145420372766 \cdot 10^{55}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{1}{2}}{x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original21.2
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -8.9999582525827406e135

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\left(-x\right) + y \cdot \frac{\frac{-1}{2}}{x}}\]

    if -8.9999582525827406e135 < x < 5.73985145420372766e55

    1. Initial program 0.0

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

    if 5.73985145420372766e55 < x

    1. Initial program 39.5

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

    2. Taylor expanded around inf 1.7

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]

    3. Simplified1.7

      \[\leadsto \color{blue}{x + y \cdot \frac{\frac{1}{2}}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.9999582525827406 \cdot 10^{135}:\\ \;\;\;\;\left(-x\right) + y \cdot \frac{\frac{-1}{2}}{x}\\ \mathbf{elif}\;x \le 5.73985145420372766 \cdot 10^{55}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{1}{2}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (neg (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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