Average Error: 33.5 → 13.0
Time: 1.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.0970310698419794 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.5510121520918392 \cdot 10^{+132}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{(\left(4 \cdot c\right) \cdot \left(-a\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{\frac{2}{c}}}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.6
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.0970310698419794e-288

    1. Initial program 21.4

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

    2. Initial simplification21.4

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

    4. Applied div-inv21.5

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

    if -5.0970310698419794e-288 < b < 1.5510121520918392e+132

    1. Initial program 33.0

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

    2. Initial simplification33.0

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

    4. Applied flip--33.1

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

    5. Applied associate-/l/37.5

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

    6. Simplified19.4

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

    8. Applied associate-/r*14.0

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

    10. Applied distribute-lft-neg-out14.0

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

    11. Applied distribute-frac-neg14.0

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

    12. Applied distribute-frac-neg14.0

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

    13. Simplified8.6

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

    14. Taylor expanded around inf 8.5

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

    if 1.5510121520918392e+132 < b

    1. Initial program 60.7

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

    2. Initial simplification60.7

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

    4. Applied flip--60.8

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

    5. Applied associate-/l/60.8

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

    6. Simplified35.2

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

    8. Applied associate-/r*34.6

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

    10. Applied distribute-lft-neg-out34.6

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

    11. Applied distribute-frac-neg34.6

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

    12. Applied distribute-frac-neg34.6

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

    13. Simplified34.3

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

    14. Taylor expanded around 0 2.5

      \[\leadsto -\frac{\frac{4}{\frac{2}{c}}}{\color{blue}{b} + b}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.0970310698419794 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.5510121520918392 \cdot 10^{+132}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{(\left(4 \cdot c\right) \cdot \left(-a\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{\frac{2}{c}}}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))