Average Error: 33.5 → 13.3
Time: 28.2s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 8.512991761568527 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b\right)}{2}\\ \mathbf{elif}\;b \le 2.9680372233693214 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{(\left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) + b)_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{b \cdot 2}}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.6
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < 8.512991761568527e-135

    1. Initial program 20.5

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

    2. Initial simplification20.5

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

    4. Applied div-inv20.6

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

    if 8.512991761568527e-135 < b < 2.9680372233693214e+129

    1. Initial program 40.9

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

    2. Initial simplification40.9

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

    4. Applied flip--41.0

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

    5. Applied associate-/l/43.8

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

    6. Simplified16.9

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

    8. Applied times-frac12.6

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

    9. Simplified5.5

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

    11. Applied associate-*r/5.4

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

    13. Applied add-sqr-sqrt5.6

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

    14. Applied fma-def5.5

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

    if 2.9680372233693214e+129 < b

    1. Initial program 60.4

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

    2. Initial simplification60.4

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

    4. Applied flip--60.5

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

    5. Applied associate-/l/60.5

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

    6. Simplified34.8

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

    8. Applied times-frac34.1

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

    9. Simplified33.8

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

    11. Applied associate-*r/33.8

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

    12. Taylor expanded around 0 2.4

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 8.512991761568527 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b\right)}{2}\\ \mathbf{elif}\;b \le 2.9680372233693214 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{(\left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) + b)_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{b \cdot 2}}{2}\\ \end{array}\]

Runtime

Time bar (total: 28.2s)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)))