Average Error: 33.6 → 9.7
Time: 3.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le -5.50549836563957 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le 4.791871306651053 \cdot 10^{-308}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le 8.869900708151351 \cdot 10^{-156}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.6
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (/ -2 2) b) < -5.50549836563957e+33

    1. Initial program 22.9

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

    3. Applied *-un-lft-identity22.9

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

    4. Applied times-frac22.8

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

    5. Applied simplify22.8

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

    if -5.50549836563957e+33 < (/ (/ -2 2) b) < 4.791871306651053e-308

    1. Initial program 54.4

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

    2. Taylor expanded around inf 45.7

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

    3. Applied simplify7.0

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

    if 4.791871306651053e-308 < (/ (/ -2 2) b) < 8.869900708151351e-156

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 2.1

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

    3. Applied simplify2.1

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

    if 8.869900708151351e-156 < (/ (/ -2 2) b)

    1. Initial program 8.4

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

    3. Applied *-un-lft-identity8.4

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

    4. Applied times-frac8.4

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

    5. Applied simplify8.4

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

  4. Applied simplify9.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le -5.50549836563957 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le 4.791871306651053 \cdot 10^{-308}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;\frac{\frac{-2}{2}}{b} \le 8.869900708151351 \cdot 10^{-156}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \end{array}}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2019053 +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)))