Average Error: 34.5 → 13.0
Time: 36.1s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.819497819135284 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 1.922455145018763 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.592377222321251 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{b} \cdot -2}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.5
Target21.3
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 4 regimes

  2. if b < -3.819497819135284e+153

    1. Initial program 60.5

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

    2. Initial simplification60.5

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

    3. Taylor expanded around -inf 2.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    4. Simplified2.2

      \[\leadsto \color{blue}{\frac{-b}{a}}\]

    if -3.819497819135284e+153 < b < 1.922455145018763e-57

    1. Initial program 13.4

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

    2. Initial simplification13.4

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

    4. Applied sub-neg13.4

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

    if 1.922455145018763e-57 < b < 1.592377222321251e+65

    1. Initial program 43.9

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

    2. Initial simplification43.9

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

    4. Applied flip--44.0

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

    5. Applied associate-/l/45.9

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

    6. Simplified17.1

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

    if 1.592377222321251e+65 < b

    1. Initial program 57.0

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

    2. Initial simplification57.0

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

    3. Taylor expanded around inf 15.0

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.819497819135284 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 1.922455145018763 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.592377222321251 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{b} \cdot -2}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 36.1s)Debug logProfile

herbie shell --seed 2018254 
(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)))