Average Error: 2.3 → 0.8
Time: 29.4s
Precision: 64
Internal Precision: 320
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.064239501953125:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < -0.064239501953125

    1. Initial program 4.1

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

    3. Applied p16-flip--3.6

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

    4. Applied associate-/l/3.7

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

    5. Simplified1.1

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

    7. Applied associate-/r*1.0

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

    if -0.064239501953125 < b

    1. Initial program 0.7

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

    3. Applied associate-/r*0.7

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(real->posit(4)\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(real->posit(2)\right)}\right)}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.064239501953125:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))