Average Error: 33.8 → 15.7
Time: 23.5s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.336244498910505 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3485817847161475 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b} \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target21.2
Herbie15.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.336244498910505e+154

    1. Initial program 60.9

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

    2. Initial simplification60.9

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

    3. Taylor expanded around -inf 52.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}}\]

    if -1.336244498910505e+154 < b < 1.3485817847161475e-49

    1. Initial program 13.0

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

    2. Initial simplification13.0

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

    4. Applied clear-num13.1

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

    6. Applied div-inv13.2

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

    if 1.3485817847161475e-49 < b

    1. Initial program 53.7

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

    2. Initial simplification53.7

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

    4. Applied clear-num53.7

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

    6. Applied div-inv53.7

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

    7. Taylor expanded around 0 8.3

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

    8. Simplified8.3

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.336244498910505 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3485817847161475 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b} \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Runtime

Time bar (total: 23.5s)Debug logProfile

herbie shell --seed 2018248 +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)))