Average Error: 33.0 → 9.8
Time: 1.2m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le -5.9036366863779675 \cdot 10^{-19}:\\ \;\;\;\;\frac{c}{\frac{b/2}{\frac{-1}{2}}}\\ \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le -4.5628751482318626 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\\ \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le -1.5369932296683007 \cdot 10^{-132}:\\ \;\;\;\;\frac{c}{\frac{b/2}{\frac{-1}{2}}}\\ \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le 105.97948941423607:\\ \;\;\;\;\left(\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (/ b/2 -1/2) < -5.9036366863779675e-19 or -4.5628751482318626e-119 < (/ b/2 -1/2) < -1.5369932296683007e-132

    1. Initial program 54.1

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 18.2

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a}\]

    3. Applied simplify6.8

      \[\leadsto \color{blue}{\frac{c}{\frac{b/2}{\frac{-1}{2}}}}\]

    if -5.9036366863779675e-19 < (/ b/2 -1/2) < -4.5628751482318626e-119

    1. Initial program 35.4

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+35.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b/2\right) \cdot \left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c} \cdot \sqrt{b/2 \cdot b/2 - a \cdot c}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}}{a}\]

    4. Applied simplify18.1

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\]

    if -1.5369932296683007e-132 < (/ b/2 -1/2) < 105.97948941423607

    1. Initial program 12.3

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv12.4

      \[\leadsto \color{blue}{\left(\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if 105.97948941423607 < (/ b/2 -1/2)

    1. Initial program 29.7

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 8.2

      \[\leadsto \color{blue}{-2 \cdot \frac{b/2}{a}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +o rules:numerics
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))