Average Error: 33.0 → 13.2
Time: 2.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.069954379081001 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \mathbf{if}\;b \le 1.677899329140992 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2} \cdot \left(\left(-c\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-4\right)}{b \cdot \left(2 + 2\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.4
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < 7.069954379081001e-295

    1. Initial program 20.7

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

    2. Applied simplify20.7

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

    if 7.069954379081001e-295 < b < 1.677899329140992e+43

    1. Initial program 28.1

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

    2. Applied simplify28.1

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

    4. Applied flip--28.2

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

    5. Applied simplify16.1

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

    7. Applied clear-num16.2

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

    9. Applied associate-/r/16.1

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

    10. Applied times-frac15.9

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

    11. Applied simplify9.1

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

    if 1.677899329140992e+43 < b

    1. Initial program 56.3

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

    2. Applied simplify56.3

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

    4. Applied flip--56.3

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

    5. Applied simplify28.8

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

    7. Applied clear-num29.0

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

    9. Applied associate-/r/28.9

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

    10. Applied times-frac26.9

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

    11. Applied simplify25.2

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

    12. Taylor expanded around 0 4.7

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

    13. Applied simplify4.6

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))