Average Error: 33.8 → 15.4
Time: 26.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.269902968546897 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.645878839556587 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \frac{\frac{1}{2}}{a}}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.6
Herbie15.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < 1.269902968546897e-131

    1. Initial program 20.9

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

    2. Initial simplification20.9

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

    3. Taylor expanded around -inf 20.9

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

    4. Simplified21.0

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

    if 1.269902968546897e-131 < b < 9.645878839556587e+129

    1. Initial program 41.3

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

    2. Initial simplification41.3

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

    3. Taylor expanded around -inf 41.3

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

    4. Simplified41.4

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

    6. Applied div-inv41.4

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

    8. Applied flip--41.5

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

    9. Applied associate-*l/41.5

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

    10. Simplified13.6

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

    if 9.645878839556587e+129 < b

    1. Initial program 60.5

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

    2. Initial simplification60.5

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

    3. Taylor expanded around -inf 60.5

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

    4. Simplified60.5

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

    6. Applied *-un-lft-identity60.5

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

    7. Applied associate-/l*60.5

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

    8. Taylor expanded around 0 2.3

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

    9. Simplified2.3

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.269902968546897 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.645878839556587 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \frac{\frac{1}{2}}{a}}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Runtime

Time bar (total: 26.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.515.412.519.084.8%
herbie shell --seed 2018353 +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)))