Average Error: 34.2 → 10.3
Time: 3.8m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.614644252084147 \cdot 10^{-35}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.180471078121508 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot c\right) \cdot a\right) \cdot \frac{1}{2}}{\left(b - \sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}\right) \cdot a}\\ \mathbf{elif}\;b \le -4.380258891361565 \cdot 10^{-166}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 5.1457556548524475 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.5
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.614644252084147e-35 or -1.180471078121508e-156 < b < -4.380258891361565e-166

    1. Initial program 53.3

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

    2. Simplified53.3

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

    4. Applied *-un-lft-identity53.3

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

    5. Applied div-inv53.3

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

    6. Applied times-frac53.3

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

    7. Simplified53.3

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

    8. Simplified53.3

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

    9. Taylor expanded around -inf 8.5

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

    10. Simplified8.5

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

    if -6.614644252084147e-35 < b < -1.180471078121508e-156

    1. Initial program 30.2

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

    2. Simplified30.2

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

    4. Applied *-un-lft-identity30.2

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

    5. Applied div-inv30.2

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

    6. Applied times-frac30.2

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

    7. Simplified30.2

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

    8. Simplified30.2

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

    10. Applied *-un-lft-identity30.2

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

    11. Applied associate-/r*30.2

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

    12. Applied flip-+30.3

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

    13. Applied distribute-neg-frac30.3

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

    14. Applied frac-times33.8

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

    15. Simplified23.7

      \[\leadsto \frac{\color{blue}{\left(0 - a \cdot \left(c \cdot -4\right)\right) \cdot \frac{-1}{2}}}{\left(b - \sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*}\right) \cdot a}\]

    if -4.380258891361565e-166 < b < 5.1457556548524475e+64

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied add-sqr-sqrt12.2

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

    if 5.1457556548524475e+64 < b

    1. Initial program 38.5

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

    2. Simplified38.5

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

    3. Taylor expanded around inf 4.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.614644252084147 \cdot 10^{-35}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.180471078121508 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot c\right) \cdot a\right) \cdot \frac{1}{2}}{\left(b - \sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}\right) \cdot a}\\ \mathbf{elif}\;b \le -4.380258891361565 \cdot 10^{-166}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 5.1457556548524475 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +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)))