Average Error: 33.3 → 6.3
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.193865508518113 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.305063165236378 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.949338867770777 \cdot 10^{+125}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.4
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.193865508518113e+143

    1. Initial program 57.8

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

    2. Taylor expanded around -inf 2.5

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

    if -6.193865508518113e+143 < b < 9.305063165236378e-253

    1. Initial program 8.5

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

    2. Taylor expanded around 0 8.5

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

    3. Simplified8.5

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

    if 9.305063165236378e-253 < b < 6.949338867770777e+125

    1. Initial program 36.2

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

    3. Applied flip-+36.3

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

    4. Applied associate-/l/39.7

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

    5. Simplified20.1

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

    7. Applied associate-/r*15.5

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

    8. Simplified15.5

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

    10. Applied *-un-lft-identity15.5

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

    11. Applied *-un-lft-identity15.5

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

    12. Applied distribute-lft-out--15.5

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

    13. Applied times-frac15.5

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

    14. Applied times-frac15.4

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

    15. Simplified15.4

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

    16. Simplified8.0

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

    18. Applied add-sqr-sqrt8.0

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

    19. Applied sqrt-prod8.2

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

    if 6.949338867770777e+125 < b

    1. Initial program 60.5

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

    2. Taylor expanded around inf 1.6

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

    3. Simplified1.6

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.193865508518113 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.305063165236378 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.949338867770777 \cdot 10^{+125}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018278 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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