Average Error: 33.8 → 8.9
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.1382247414568033 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le 1.0297431311884128 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{elif}\;b \le 2.761124983263226 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{c}{\frac{-1}{2}}}{b + \sqrt{(b \cdot b + \left(\left(-4 \cdot c\right) \cdot a\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.6
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1382247414568033e+125

    1. Initial program 51.4

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

    2. Simplified51.4

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

    3. Taylor expanded around -inf 8.8

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

    4. Simplified2.8

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

    if -3.1382247414568033e+125 < b < 1.0297431311884128e-307

    1. Initial program 8.5

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

    2. Simplified8.5

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

    3. Taylor expanded around inf 8.5

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

    4. Simplified8.5

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

    if 1.0297431311884128e-307 < b < 2.761124983263226e+151

    1. Initial program 35.0

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

    2. Simplified35.0

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

    4. Applied flip--35.1

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

    5. Applied associate-/l/35.1

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

    6. Simplified15.8

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

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

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

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

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

    10. Applied times-frac15.8

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

    11. Simplified15.8

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

    12. Simplified8.0

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

    if 2.761124983263226e+151 < b

    1. Initial program 62.6

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

    2. Simplified62.6

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

    3. Taylor expanded around inf 13.8

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

    4. Simplified15.9

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.1382247414568033 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le 1.0297431311884128 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{elif}\;b \le 2.761124983263226 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{c}{\frac{-1}{2}}}{b + \sqrt{(b \cdot b + \left(\left(-4 \cdot c\right) \cdot a\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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)))