Average Error: 33.7 → 12.8
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3395204081552134 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -1.96292444197926 \cdot 10^{-264}:\\ \;\;\;\;\left(\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.473326279994868 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target21.0
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3395204081552134e+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 -1.3395204081552134e+154 < b < -1.96292444197926e-264

    1. Initial program 8.3

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

    2. Initial simplification8.3

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

    4. Applied div-inv8.5

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

    if -1.96292444197926e-264 < b < 1.473326279994868e+84

    1. Initial program 30.9

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

    2. Initial simplification30.9

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

    4. Applied flip--31.1

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

    5. Applied associate-/l/35.7

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

    6. Simplified22.1

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

    8. Applied associate-/r*16.7

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

    9. Taylor expanded around 0 10.3

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

    10. Taylor expanded around 0 10.3

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

    if 1.473326279994868e+84 < b

    1. Initial program 57.8

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

    2. Initial simplification57.8

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

    4. Applied flip--57.9

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

    5. Applied associate-/l/58.2

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

    6. Simplified30.7

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

    8. Applied associate-/r*29.0

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

    9. Taylor expanded around 0 27.9

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

    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 simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3395204081552134 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le -1.96292444197926 \cdot 10^{-264}:\\ \;\;\;\;\left(\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.473326279994868 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018225 +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)))