Average Error: 33.5 → 6.7
Time: 2.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7581975230643 \cdot 10^{+93}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -1.3023743998422597 \cdot 10^{-272}:\\ \;\;\;\;\left(\frac{c}{2} \cdot 4\right) \cdot \frac{1}{\sqrt{(\left(c \cdot a\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}\\ \mathbf{if}\;b \le 1.1540544437409996 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.2
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 < -1.7581975230643e+93

    1. Initial program 58.8

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

    2. Taylor expanded around -inf 40.6

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

    3. Applied simplify2.6

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

    if -1.7581975230643e+93 < b < -1.3023743998422597e-272

    1. Initial program 32.9

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

    3. Applied flip--33.0

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

    4. Applied simplify16.1

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

    5. Applied simplify16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac14.6

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

    9. Applied times-frac9.4

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

    10. Applied simplify9.4

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

    11. Applied simplify8.3

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

    if -1.3023743998422597e-272 < b < 1.1540544437409996e+66

    1. Initial program 9.6

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

    3. Applied add-cube-cbrt10.4

      \[\leadsto \frac{\left(-b\right) - \color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Applied add-cube-cbrt10.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - \left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Applied prod-diff10.7

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

    6. Applied simplify9.7

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

    7. Applied simplify9.6

      \[\leadsto \frac{\left(\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}\right) + \color{blue}{0}}{2 \cdot a}\]

    if 1.1540544437409996e+66 < b

    1. Initial program 38.1

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

    2. Taylor expanded around inf 10.2

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

    3. Applied simplify5.2

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

  4. Applied simplify6.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.7581975230643 \cdot 10^{+93}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -1.3023743998422597 \cdot 10^{-272}:\\ \;\;\;\;\left(\frac{c}{2} \cdot 4\right) \cdot \frac{1}{\sqrt{(\left(c \cdot a\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}\\ \mathbf{if}\;b \le 1.1540544437409996 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018178 +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)))