Average Error: 37.3 → 8.1
Time: 1.1m
Precision: 64
Ground Truth: 128
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -4.531827883994214 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{c}{2} \cdot 4}{\frac{c + c}{\frac{b}{a}} - \left(b - \left(-b\right)\right)}\\ \mathbf{if}\;b \le 1438490.0801097094:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original37.3
Comparison25.2
Herbie8.1
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if b < -4.531827883994214e-14

    1. Initial program 58.4

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

    3. Applied flip-- 58.4

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

    4. Applied simplify 32.7

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

    5. Applied taylor 15.5

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

    6. Taylor expanded around -inf 15.5

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

    7. Applied simplify 2.6

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

    8. Applied simplify 2.6

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

    if -4.531827883994214e-14 < b < 1438490.0801097094

    1. Initial program 18.1

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

    3. Applied div-inv 18.2

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

    if 1438490.0801097094 < b

    1. Initial program 37.8

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

    2. Applied taylor 10.0

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

    3. Taylor expanded around inf 10.0

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

    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 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 1.1m Debug log

Please include this information when filing a bug report:

herbie --seed '#(4025883511 1929518346 2149116964 2318303697 2782024243 2846644129)'
(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)))