Average Error: 33.5 → 6.4
Time: 41.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.084878034205583 \cdot 10^{+132}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.880353190350159 \cdot 10^{-264}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 1.890501396456425 \cdot 10^{+92}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.7
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.084878034205583e+132

    1. Initial program 61.0

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

    2. Initial simplification61.0

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

    3. Taylor expanded around -inf 2.0

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

    4. Simplified2.0

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

    if -4.084878034205583e+132 < b < -2.880353190350159e-264

    1. Initial program 35.8

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

    2. Initial simplification35.9

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

    4. Applied flip--36.0

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

    5. Applied associate-/l/39.5

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

    6. Simplified19.8

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

    8. Applied associate-/r*14.8

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

    9. Simplified14.7

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

    10. Taylor expanded around -inf 7.6

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

    if -2.880353190350159e-264 < b < 1.890501396456425e+92

    1. Initial program 9.1

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

    2. Initial simplification9.1

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

    3. Taylor expanded around -inf 9.1

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

    4. Simplified9.1

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

    if 1.890501396456425e+92 < b

    1. Initial program 44.7

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

    2. Initial simplification44.7

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

    4. Applied flip--61.2

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

    5. Applied associate-/l/61.7

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

    6. Simplified61.8

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

    8. Applied associate-/r*61.3

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

    9. Simplified61.3

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

    10. Taylor expanded around -inf 61.2

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

    11. Taylor expanded around 0 4.1

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

    12. Simplified4.1

      \[\leadsto \color{blue}{\frac{-b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.084878034205583 \cdot 10^{+132}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.880353190350159 \cdot 10^{-264}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 1.890501396456425 \cdot 10^{+92}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Runtime

Time bar (total: 41.1s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))