Average Error: 33.6 → 6.9
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -4.145680582588426 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 2.8010928323165732 \cdot 10^{-297}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{if}\;-b \le 1.1872976266102309 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}}}{2} \cdot \frac{1}{\sqrt{\sqrt{(\left(a \cdot c\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.7
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -4.145680582588426e+97

    1. Initial program 44.2

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

    2. Taylor expanded around inf 10.5

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

    3. Applied simplify4.3

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

    if -4.145680582588426e+97 < (- b) < 2.8010928323165732e-297

    1. Initial program 9.0

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

    3. Applied clear-num9.1

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

    if 2.8010928323165732e-297 < (- b) < 1.1872976266102309e+118

    1. Initial program 33.9

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

    3. Applied flip--34.0

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

    4. Applied simplify16.6

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

    5. Applied simplify16.6

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

    7. Applied add-sqr-sqrt16.8

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

    8. Applied times-frac14.7

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

    9. Applied times-frac9.7

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

    10. Applied simplify9.0

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

    if 1.1872976266102309e+118 < (- b)

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 40.3

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

    3. Applied simplify2.5

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

  4. Applied simplify6.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -4.145680582588426 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 2.8010928323165732 \cdot 10^{-297}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{if}\;-b \le 1.1872976266102309 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}}}{2} \cdot \frac{1}{\sqrt{\sqrt{(\left(a \cdot c\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))