Average Error: 33.7 → 8.6
Time: 1.6m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3907652222998155 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\ \mathbf{if}\;b \le 1.3192368517509897 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\frac{a + a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}}\\ \mathbf{if}\;b \le 6.578629845313264 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c + c}{\frac{b}{a}} - \left(b + b\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.6
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3907652222998155e+110

    1. Initial program 46.4

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

    2. Taylor expanded around -inf 8.4

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

    3. Applied simplify2.5

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

    if -1.3907652222998155e+110 < b < 1.3192368517509897e-110

    1. Initial program 11.7

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

    3. Applied clear-num11.9

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

    if 1.3192368517509897e-110 < b < 6.578629845313264e+85

    1. Initial program 41.1

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

    3. Applied flip-+41.2

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

    4. Applied simplify14.9

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

    if 6.578629845313264e+85 < b

    1. Initial program 58.4

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

    3. Applied flip-+58.4

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

    4. Applied simplify30.8

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

    5. Taylor expanded around inf 14.2

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

    6. Applied simplify2.4

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

  4. Applied simplify8.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.3907652222998155 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\ \mathbf{if}\;b \le 1.3192368517509897 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\frac{a + a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}}\\ \mathbf{if}\;b \le 6.578629845313264 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c + c}{\frac{b}{a}} - \left(b + b\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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)))