Average Error: 33.5 → 9.7
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.663483308794079 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 1.6168433953466018 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2 \cdot 2}}{\frac{c}{\frac{b}{a}} - b}\\ \end{array}\]

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

Original33.5
Target20.5
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.663483308794079e+98

    1. Initial program 44.9

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

    2. Taylor expanded around -inf 11.2

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

    3. Applied simplify4.6

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}}\]

    if -2.663483308794079e+98 < b < 1.6168433953466018e-55

    1. Initial program 13.0

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

    if 1.6168433953466018e-55 < b

    1. Initial program 53.5

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

    3. Applied flip-+53.5

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

    4. Applied simplify25.6

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

    6. Applied *-un-lft-identity25.6

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

    7. Applied times-frac25.6

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

    8. Applied simplify20.4

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

    9. Taylor expanded around inf 10.0

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

    10. Applied simplify7.9

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

  4. Applied simplify9.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -2.663483308794079 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 1.6168433953466018 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2 \cdot 2}}{\frac{c}{\frac{b}{a}} - b}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018170 
(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)))