Average Error: 33.9 → 6.7
Time: 1.6m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(-b\right) - b \le -1.602744556913015 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;\left(-b\right) - b \le 6.8772714888223525 \cdot 10^{-236}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{if}\;\left(-b\right) - b \le 5.593927708783694 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{(\left(\sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*}\right) \cdot \left(\frac{\frac{1}{2}}{c}\right) + \left(b \cdot \frac{\frac{-1}{2}}{c}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.9
Target20.7
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (- b) b) < -1.602744556913015e+73

    1. Initial program 40.2

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

    2. Applied simplify40.1

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

    3. Taylor expanded around 0 3.8

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

    if -1.602744556913015e+73 < (- (- b) b) < 6.8772714888223525e-236

    1. Initial program 10.2

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

    2. Applied simplify10.2

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

    4. Applied div-inv10.3

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

    5. Applied simplify10.3

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

    if 6.8772714888223525e-236 < (- (- b) b) < 5.593927708783694e+147

    1. Initial program 37.0

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

    2. Applied simplify37.0

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

    4. Applied flip--37.1

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

    5. Applied simplify16.4

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

    6. Applied simplify16.4

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

    8. Applied clear-num16.6

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

    9. Applied simplify7.5

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

    if 5.593927708783694e+147 < (- (- b) b)

    1. Initial program 62.1

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

    2. Applied simplify62.1

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

    3. Taylor expanded around -inf 14.2

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

    4. Applied simplify1.9

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

  4. Applied simplify6.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(-b\right) - b \le -1.602744556913015 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;\left(-b\right) - b \le 6.8772714888223525 \cdot 10^{-236}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{if}\;\left(-b\right) - b \le 5.593927708783694 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{(\left(\sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*}\right) \cdot \left(\frac{\frac{1}{2}}{c}\right) + \left(b \cdot \frac{\frac{-1}{2}}{c}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))