Average Error: 33.6 → 12.4
Time: 1.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.344410575089653 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -3.1599935033480256 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot \left(-4\right) + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 7.62762473528317 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target21.1
Herbie12.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.344410575089653e+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

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

    3. Taylor expanded around -inf 52.2

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

    if -1.344410575089653e+154 < b < -3.1599935033480256e-305

    1. Initial program 8.8

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

    2. Initial simplification8.8

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

    4. Applied fma-udef8.8

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

    if -3.1599935033480256e-305 < b < 7.62762473528317e+85

    1. Initial program 31.7

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

    2. Initial simplification31.7

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

    4. Applied flip--31.9

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

    5. Applied associate-/l/36.6

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

    6. Simplified21.9

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

    8. Applied associate-/r*16.5

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

    9. Taylor expanded around -inf 9.2

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

    if 7.62762473528317e+85 < b

    1. Initial program 57.6

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

    2. Initial simplification57.6

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

    4. Applied flip--57.6

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

    5. Applied associate-/l/57.8

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

    6. Simplified32.1

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

    8. Applied associate-/r*30.5

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

    9. Taylor expanded around -inf 29.4

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

    10. Taylor expanded around 0 3.2

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{b} + b}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.344410575089653 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -3.1599935033480256 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot \left(-4\right) + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 7.62762473528317 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018242 +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)))