Average Error: 33.1 → 8.1
Time: 38.9s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.76223481168814 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.1097631082543147 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1.645760677557775 \cdot 10^{+50}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \left(b - \frac{a \cdot c}{b} \cdot 2\right)\right)}\\ \end{array}\]

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

Original33.1
Target20.3
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.76223481168814e+153

    1. Initial program 60.7

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -6.76223481168814e+153 < b < -2.1097631082543147e-305

    1. Initial program 8.7

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

    3. Applied clear-num8.9

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

    if -2.1097631082543147e-305 < b < 1.645760677557775e+50

    1. Initial program 29.5

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

    3. Applied flip-+29.6

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

    4. Applied associate-/l/34.6

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

    5. Simplified22.5

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

    7. Applied times-frac17.0

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

    8. Simplified9.8

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

    10. Applied frac-times9.7

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

    if 1.645760677557775e+50 < b

    1. Initial program 55.7

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

    3. Applied flip-+55.7

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

    4. Applied associate-/l/56.4

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

    5. Simplified28.1

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

    7. Applied times-frac26.4

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

    8. Simplified25.0

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

    10. Applied frac-times25.0

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

    11. Taylor expanded around inf 7.7

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.76223481168814 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.1097631082543147 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1.645760677557775 \cdot 10^{+50}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \left(b - \frac{a \cdot c}{b} \cdot 2\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 38.9s)Debug logProfile

herbie shell --seed 2018219 
(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)))