Average Error: 32.8 → 9.2
Time: 1.8m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c \cdot \frac{4}{-2}}{\frac{\frac{c}{b}}{1}} \le -1.1196487313367502 \cdot 10^{+102}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \mathbf{if}\;\frac{c \cdot \frac{4}{-2}}{\frac{\frac{c}{b}}{1}} \le -1.2092469738969913 \cdot 10^{-241}:\\ \;\;\;\;\frac{\frac{c \cdot a}{1} \cdot \frac{4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{if}\;\frac{c \cdot \frac{4}{-2}}{\frac{\frac{c}{b}}{1}} \le 1.7241096223835888 \cdot 10^{+65}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.8
Target20.4
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (* c (/ 4 -2)) (/ (/ c b) 1)) < -1.1196487313367502e+102

    1. Initial program 56.5

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

    2. Taylor expanded around inf 13.4

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

    3. Applied simplify2.4

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

    if -1.1196487313367502e+102 < (/ (* c (/ 4 -2)) (/ (/ c b) 1)) < -1.2092469738969913e-241

    1. Initial program 36.2

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

    3. Applied flip-+36.3

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

    4. Applied simplify16.2

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

    6. Applied *-un-lft-identity16.2

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

    7. Applied times-frac16.3

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

    if -1.2092469738969913e-241 < (/ (* c (/ 4 -2)) (/ (/ c b) 1)) < 1.7241096223835888e+65

    1. Initial program 11.3

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

    3. Applied add-cube-cbrt11.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    4. Applied fma-def11.6

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right))_*}}{2 \cdot a}\]

    if 1.7241096223835888e+65 < (/ (* c (/ 4 -2)) (/ (/ c b) 1))

    1. Initial program 38.1

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

    2. Taylor expanded around -inf 9.2

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

    3. Applied simplify4.6

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(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)))