Average Error: 33.7 → 6.9
Time: 42.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1260631204365375 \cdot 10^{+153}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -3.7398581097526083 \cdot 10^{-290}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.7
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1260631204365375e+153

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 1.4

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

    3. Simplified1.4

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

    if -1.1260631204365375e+153 < b < -3.7398581097526083e-290

    1. Initial program 34.3

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

    2. Taylor expanded around 0 34.3

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

    3. Simplified34.3

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

    5. Applied div-inv34.3

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot -4\right) \cdot a}\right) \cdot \frac{1}{2 \cdot a}}\]
    6. Using strategy rm

    7. Applied flip--34.4

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

    8. Applied associate-*l/34.5

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

    9. Simplified14.0

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

    10. Taylor expanded around 0 7.1

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

    if -3.7398581097526083e-290 < b < 1.7760913692530284e+28

    1. Initial program 10.9

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

    2. Taylor expanded around 0 10.9

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

    3. Simplified10.9

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

    if 1.7760913692530284e+28 < b

    1. Initial program 33.8

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

    2. Taylor expanded around 0 33.8

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

    3. Simplified33.8

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

    5. Applied div-inv33.9

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

    6. Taylor expanded around inf 6.8

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1260631204365375 \cdot 10^{+153}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -3.7398581097526083 \cdot 10^{-290}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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