Average Error: 37.3 → 8.1

Time: 1.1m

Precision: 64

Ground Truth: 128

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

⬇

\[\begin{array}{l} \mathbf{if}\;b \le -4.531827883994214 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c \cdot 2}{\frac{b}{a}} - b \cdot 2}\\ \mathbf{if}\;b \le 1438490.0801097094:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

The X axis may use a short exponential scale — Bits error versus `a`

Target

Original	37.3
Comparison	25.2
Herbie	8.1

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

Derivation

Split input into 3 regimes.
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
  \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
6. Taylor expanded around -inf 15.5
  
  \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}}{2 \cdot a}\]
7. Applied simplify 2.6
  \[\leadsto \color{blue}{\frac{\frac{1}{\frac{2}{4}} \cdot c}{\frac{c \cdot 2}{\frac{b}{a}} - b \cdot 2}}\]
8. Applied simplify 2.6
  \[\leadsto \frac{\color{blue}{\frac{4}{2} \cdot c}}{\frac{c \cdot 2}{\frac{b}{a}} - b \cdot 2}\]
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
  \[\leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
Recombined 3 regimes into one program.
Removed slow pow expressions

Runtime

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)))

Error

Target

Derivation

`if b < -4.531827883994214e-14`

`if -4.531827883994214e-14 < b < 1438490.0801097094`

`if 1438490.0801097094 < b`

Runtime