Average Error: 33.8 → 4.9
Time: 52.8s
Precision: 64
Ground Truth: 128
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -4.3723578359122876 \cdot 10^{+151}:\\ \;\;\;\;\frac{-2}{2} \cdot \frac{c}{b}\\ \mathbf{if}\;b \le 7.670790017984618 \cdot 10^{-300}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}\\ \mathbf{if}\;b \le 3.098843606291808 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Comparison21.2
Herbie4.9
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes.

  2. if b < -4.3723578359122876e+151

    1. Initial program 62.6

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Applied taylor 14.8

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

    3. Taylor expanded around -inf 14.8

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

    4. Applied simplify 0

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

    if -4.3723578359122876e+151 < b < 7.670790017984618e-300

    1. Initial program 33.2

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-- 33.3

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Applied simplify 15.4

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity 15.4

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

    7. Applied times-frac 15.4

      \[\leadsto \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    8. Applied times-frac 15.4

      \[\leadsto \color{blue}{\frac{\frac{4}{1}}{2} \cdot \frac{\frac{a \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]

    9. Applied simplify 15.4

      \[\leadsto \color{blue}{\frac{4}{2}} \cdot \frac{\frac{a \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]

    10. Applied simplify 7.8

      \[\leadsto \frac{4}{2} \cdot \color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}\]

    if 7.670790017984618e-300 < b < 3.098843606291808e+59

    1. Initial program 7.9

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv 8.1

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 3.098843606291808e+59 < b

    1. Initial program 41.1

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Applied taylor 12.3

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

    3. Taylor expanded around inf 12.3

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

    4. Applied simplify 0.0

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

    5. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 52.8s Debug log

Please report a bug with the following info:

herbie --seed '#(3847065007 2273462729 3382707546 1267762998 3508967231 1546490868)'
(FPCore (a b c)
  :name "NMSE p42, negative"

  :target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))