Average Error: 33.9 → 8.9
Time: 2.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -1.3645369122692468 \cdot 10^{+59}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot \left(2 \cdot c\right) + \left(\left(-b\right) - b\right))_*}{a \cdot 2}\\ \mathbf{if}\;-b \le 4.88870533228829 \cdot 10^{-274}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{if}\;-b \le 1.453025480714704 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{a \cdot 4}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}} \cdot \frac{c}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{2}{c}}}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.9
Target21.1
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -1.3645369122692468e+59

    1. Initial program 37.7

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

    2. Taylor expanded around inf 10.2

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

    3. Applied simplify4.8

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

    if -1.3645369122692468e+59 < (- b) < 4.88870533228829e-274

    1. Initial program 10.4

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

    3. Applied div-inv10.5

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

    if 4.88870533228829e-274 < (- b) < 1.453025480714704e+127

    1. Initial program 34.3

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

    3. Applied flip--34.4

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

    4. Applied simplify16.1

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

    5. Applied simplify16.1

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

    7. Applied add-sqr-sqrt16.2

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

    8. Applied times-frac14.7

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

    if 1.453025480714704e+127 < (- b)

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 13.7

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

    3. Applied simplify1.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{1}}{\frac{2}{c}}}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify8.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -1.3645369122692468 \cdot 10^{+59}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot \left(2 \cdot c\right) + \left(\left(-b\right) - b\right))_*}{a \cdot 2}\\ \mathbf{if}\;-b \le 4.88870533228829 \cdot 10^{-274}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{if}\;-b \le 1.453025480714704 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{a \cdot 4}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}} \cdot \frac{c}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{2}{c}}}{b}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2020178 +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)))