Average Error: 33.4 → 13.1
Time: 3.8m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.286013373950679 \cdot 10^{+115}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;b \le 1.5130832338357297 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{if}\;b \le 1.293965168376474 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{a + a}{\frac{b}{c}} + \left(\left(-b\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{a + a}{\frac{b}{c}} + \left(\left(-b\right) - b\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.7
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.286013373950679e+115

    1. Initial program 48.2

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

    2. Taylor expanded around -inf 10.1

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

    3. Applied simplify3.7

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

    4. Applied simplify3.7

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

    if -4.286013373950679e+115 < b < 1.5130832338357297e-246

    1. Initial program 10.0

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

    3. Applied div-inv10.1

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

    if 1.5130832338357297e-246 < b < 1.293965168376474e+84

    1. Initial program 32.6

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

    3. Applied flip-+32.7

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

    4. Applied simplify15.9

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

    6. Applied clear-num16.1

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

    7. Applied simplify8.3

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

    8. Taylor expanded around inf 33.3

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

    9. Applied simplify32.8

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

    if 1.293965168376474e+84 < b

    1. Initial program 57.9

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

    3. Applied flip-+58.0

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

    4. Applied simplify31.2

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

    6. Applied clear-num31.4

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

    7. Applied simplify28.9

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

    8. Taylor expanded around inf 7.7

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

    9. Applied simplify2.9

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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)))