Average Error: 33.5 → 22.5
Time: 17.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6.04020777315113 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \frac{1}{a \cdot 2} + \frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.2
Herbie22.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 6.04020777315113e-133

    1. Initial program 20.4

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

    2. Simplified20.4

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

    4. Applied *-un-lft-identity20.4

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

    5. Applied associate-/l*20.5

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

    7. Applied associate-/r/20.5

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

    9. Applied sub-neg20.5

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

    10. Applied distribute-rgt-in20.5

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

    11. Simplified20.5

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

    if 6.04020777315113e-133 < b

    1. Initial program 50.2

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

    2. Simplified50.2

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

    4. Applied *-un-lft-identity50.2

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

    5. Applied associate-/l*50.2

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

    7. Applied associate-/r/50.2

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

    9. Applied flip--50.3

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

    10. Applied frac-times51.8

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

    11. Simplified25.1

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.04020777315113 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \frac{1}{a \cdot 2} + \frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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)))