Average Error: 33.7 → 14.1
Time: 20.2s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.4397918726816674 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.5682435268134212 \cdot 10^{+154}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \frac{\frac{1}{2}}{a}}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.7
Herbie14.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.4397918726816674e-306

    1. Initial program 21.2

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

    2. Initial simplification21.2

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

    3. Taylor expanded around 0 21.2

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

    4. Simplified21.2

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

    if -4.4397918726816674e-306 < b < 2.5682435268134212e+154

    1. Initial program 34.8

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

    2. Initial simplification34.8

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

    4. Applied div-inv34.8

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

    6. Applied flip--34.9

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

    7. Applied associate-*l/35.0

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

    8. Simplified14.5

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

    9. Taylor expanded around 0 8.8

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

    if 2.5682435268134212e+154 < b

    1. Initial program 62.9

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

    2. Initial simplification62.9

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

    4. Applied div-inv62.9

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

    6. Applied flip--62.9

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

    7. Applied associate-*l/62.9

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

    8. Simplified38.1

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

    9. Taylor expanded around 0 7.4

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.4397918726816674 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.5682435268134212 \cdot 10^{+154}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \frac{\frac{1}{2}}{a}}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 20.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.414.114.015.599.2%
herbie shell --seed 2018274 +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)))