Average Error: 33.8 → 12.8
Time: 33.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.952898572330902 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)\\ \mathbf{elif}\;b \le 8.009407322407041 \cdot 10^{+141}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -9.952898572330902e-304

    1. Initial program 21.1

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

    2. Initial simplification21.1

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

    4. Applied div-inv21.2

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

    if -9.952898572330902e-304 < b < 8.009407322407041e+141

    1. Initial program 34.5

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

    2. Initial simplification34.5

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

    4. Applied div-inv34.5

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

    6. Applied flip--34.6

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

    7. Applied associate-*l/34.6

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

    8. Simplified14.9

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

    9. Taylor expanded around 0 8.3

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

    if 8.009407322407041e+141 < b

    1. Initial program 61.5

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

    2. Initial simplification61.5

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

    4. Applied div-inv61.5

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

    6. Applied flip--61.5

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

    7. Applied associate-*l/61.5

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

    8. Simplified36.2

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

    9. Taylor expanded around 0 35.9

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

    10. Taylor expanded around 0 1.8

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.952898572330902 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)\\ \mathbf{elif}\;b \le 8.009407322407041 \cdot 10^{+141}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 33.1s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))