Average Error: 33.1 → 9.3
Time: 33.7s
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.932106039850163 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.903986414186807 \cdot 10^{-107}:\\ \;\;\;\;(b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\right))_*\\ \mathbf{elif}\;b \le 2.3901785402199755 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot -4}{\left(b + \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.2
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.932106039850163e+118

    1. Initial program 48.6

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

    2. Initial simplification48.6

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

    4. Applied div-sub48.6

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

    6. Applied div-inv48.6

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

    7. Applied add-cube-cbrt48.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}} - b \cdot \frac{1}{2 \cdot a}\]

    8. Applied prod-diff48.8

      \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) + \left(-\frac{1}{2 \cdot a} \cdot b\right))_* + (\left(-\frac{1}{2 \cdot a}\right) \cdot b + \left(\frac{1}{2 \cdot a} \cdot b\right))_*}\]

    9. Simplified48.6

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

    10. Simplified48.6

      \[\leadsto (b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\right))_* + \color{blue}{0}\]

    11. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right)} + 0\]

    if -4.932106039850163e+118 < b < 1.903986414186807e-107

    1. Initial program 11.2

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

    2. Initial simplification11.2

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

    4. Applied div-sub11.2

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

    6. Applied div-inv11.3

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

    7. Applied add-cube-cbrt12.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}} - b \cdot \frac{1}{2 \cdot a}\]

    8. Applied prod-diff12.0

      \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) + \left(-\frac{1}{2 \cdot a} \cdot b\right))_* + (\left(-\frac{1}{2 \cdot a}\right) \cdot b + \left(\frac{1}{2 \cdot a} \cdot b\right))_*}\]

    9. Simplified11.3

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

    10. Simplified11.3

      \[\leadsto (b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\right))_* + \color{blue}{0}\]

    if 1.903986414186807e-107 < b < 2.3901785402199755e+89

    1. Initial program 41.1

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

    2. Initial simplification41.1

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

    4. Applied flip--41.2

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

    5. Applied associate-/l/44.1

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

    6. Simplified19.4

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

    if 2.3901785402199755e+89 < b

    1. Initial program 58.1

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

    2. Initial simplification58.1

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

    4. Applied div-sub58.9

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

    6. Applied div-inv59.8

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

    7. Applied add-cube-cbrt61.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}} - b \cdot \frac{1}{2 \cdot a}\]

    8. Applied prod-diff62.1

      \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2 \cdot a}}\right) + \left(-\frac{1}{2 \cdot a} \cdot b\right))_* + (\left(-\frac{1}{2 \cdot a}\right) \cdot b + \left(\frac{1}{2 \cdot a} \cdot b\right))_*}\]

    9. Simplified62.1

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

    10. Simplified62.1

      \[\leadsto (b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\right))_* + \color{blue}{0}\]

    11. Taylor expanded around inf 2.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} + 0\]

    12. Simplified2.6

      \[\leadsto \color{blue}{\frac{-c}{b}} + 0\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.932106039850163 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.903986414186807 \cdot 10^{-107}:\\ \;\;\;\;(b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\right))_*\\ \mathbf{elif}\;b \le 2.3901785402199755 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot -4}{\left(b + \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 33.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes34.79.35.529.286.9%
herbie shell --seed 2018286 +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)))