Average Error: 34.0 → 7.1
Time: 1.8m
Precision: 64
Ground Truth: 128
\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -9.030792638214713 \cdot 10^{+68}:\\ \;\;\;\;\frac{c}{\frac{b/2}{\frac{1}{2}}} - \frac{b/2 + b/2}{a}\\ \mathbf{if}\;b/2 \le 1.6390442632966411 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;b/2 \le 3.474250069101817 \cdot 10^{+81}:\\ \;\;\;\;\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b/2 + \left(-b/2\right)}{a} - \frac{\frac{1}{2} \cdot c}{b/2}\\ \end{array}\]

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 < -9.030792638214713e+68

    1. Initial program 40.6

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

    2. Applied taylor 10.1

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

    3. Taylor expanded around -inf 10.1

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

    4. Applied simplify 0.1

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

    if -9.030792638214713e+68 < b/2 < 1.6390442632966411e-150

    1. Initial program 11.2

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

    3. Applied div-inv 11.3

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

    if 1.6390442632966411e-150 < b/2 < 3.474250069101817e+81

    1. Initial program 37.0

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

    3. Applied flip-+ 37.1

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

    4. Applied simplify 15.7

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied div-inv 15.8

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

    if 3.474250069101817e+81 < b/2

    1. Initial program 58.5

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

    2. Applied taylor 41.1

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

    3. Taylor expanded around inf 41.1

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

    4. Applied simplify 0

      \[\leadsto \color{blue}{\frac{b/2 + \left(-b/2\right)}{a} - \frac{\frac{1}{2} \cdot c}{b/2}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 1.8m Debug log

Please include this information when filing a bug report:

herbie --seed '#(1792286330 1065197884 1348530281 2149574692 2623466832 6674770)'
(FPCore (a b/2 c)
  :name "NMSE problem 3.2.1, positive"
  (/ (+ (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))