Average Error: 33.6 → 6.6
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -1.078587196574588 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{2 \cdot 2}}{\frac{c}{\frac{b}{a}} - b}\\ \mathbf{if}\;-b \le -2.8432650981668703 \cdot 10^{-263}:\\ \;\;\;\;\frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \frac{1}{2}\\ \mathbf{if}\;-b \le 1.3499173163480783 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot c}{\frac{b}{a}} - \left(b + b\right)}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.9
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -1.078587196574588e+136

    1. Initial program 61.5

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

    3. Applied flip-+61.5

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

    4. Applied simplify36.1

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

    6. Applied *-un-lft-identity36.1

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

    7. Applied times-frac36.1

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

    8. Applied simplify35.3

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

    9. Taylor expanded around inf 6.7

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

    10. Applied simplify2.0

      \[\leadsto \color{blue}{\frac{\frac{c \cdot 4}{2 \cdot 2}}{\frac{c}{\frac{b}{a}} - b}}\]

    if -1.078587196574588e+136 < (- b) < -2.8432650981668703e-263

    1. Initial program 35.3

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

    3. Applied flip-+35.4

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

    4. Applied simplify15.8

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

    6. Applied *-un-lft-identity15.8

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

    7. Applied times-frac15.8

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

    8. Applied simplify7.5

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

    if -2.8432650981668703e-263 < (- b) < 1.3499173163480783e+79

    1. Initial program 9.7

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

    3. Applied clear-num9.9

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

    4. Applied simplify9.8

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

    if 1.3499173163480783e+79 < (- b)

    1. Initial program 41.5

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

    3. Applied clear-num41.6

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

    4. Applied simplify41.6

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

    5. Taylor expanded around -inf 10.3

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

    6. Applied simplify4.6

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

  4. Applied simplify6.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -1.078587196574588 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{2 \cdot 2}}{\frac{c}{\frac{b}{a}} - b}\\ \mathbf{if}\;-b \le -2.8432650981668703 \cdot 10^{-263}:\\ \;\;\;\;\frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \frac{1}{2}\\ \mathbf{if}\;-b \le 1.3499173163480783 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot c}{\frac{b}{a}} - \left(b + b\right)}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(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)))