Average Error: 32.8 → 8.9
Time: 1.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2099900137201492 \cdot 10^{+151}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{\frac{a \cdot c}{b} \cdot 2}\\ \mathbf{elif}\;b \le 3.918269354305911 \cdot 10^{-232}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{elif}\;b \le 2.0482371071916228 \cdot 10^{+48}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.8
Target20.3
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2099900137201492e+151

    1. Initial program 60.0

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

    2. Initial simplification59.9

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

    4. Applied flip--62.2

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

    5. Applied associate-/l/62.2

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

    6. Simplified62.5

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

    8. Applied distribute-lft-neg-out62.5

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

    9. Applied distribute-frac-neg62.5

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

    10. Simplified62.4

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

    11. Taylor expanded around -inf 22.0

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

    if -1.2099900137201492e+151 < b < 3.918269354305911e-232

    1. Initial program 9.3

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

    2. Initial simplification9.3

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

    4. Applied clear-num9.5

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

    if 3.918269354305911e-232 < b < 2.0482371071916228e+48

    1. Initial program 30.3

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

    2. Initial simplification30.3

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

    4. Applied flip--30.4

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

    5. Applied associate-/l/35.4

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

    6. Simplified20.8

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

    8. Applied distribute-lft-neg-out20.8

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

    9. Applied distribute-frac-neg20.8

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

    10. Simplified7.1

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

    if 2.0482371071916228e+48 < b

    1. Initial program 56.1

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

    2. Initial simplification56.2

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

    4. Applied flip--56.2

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

    5. Applied associate-/l/56.8

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

    6. Simplified29.2

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

    8. Applied distribute-lft-neg-out29.2

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

    9. Applied distribute-frac-neg29.2

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

    10. Simplified25.9

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

    11. Taylor expanded around 0 4.4

      \[\leadsto -\frac{\left(c \cdot 4\right) \cdot \frac{1}{2}}{b + \color{blue}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2099900137201492 \cdot 10^{+151}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{\frac{a \cdot c}{b} \cdot 2}\\ \mathbf{elif}\;b \le 3.918269354305911 \cdot 10^{-232}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{elif}\;b \le 2.0482371071916228 \cdot 10^{+48}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{1}{2} \cdot \left(c \cdot 4\right)}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018217 +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)))