Average Error: 33.3 → 6.8
Time: 32.1s
Precision: 64
Internal Precision: 3200
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -3.8194526336608277 \cdot 10^{+67}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le -1.9881359338705715 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}{c}}\\ \mathbf{if}\;b/2 \le 2.104482131095671 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b/2 < -3.8194526336608277e+67

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 13.7

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a}\]

    3. Applied simplify3.2

      \[\leadsto \color{blue}{\frac{c}{b/2} \cdot \frac{-1}{2}}\]

    if -3.8194526336608277e+67 < b/2 < -1.9881359338705715e-231

    1. Initial program 34.6

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

    3. Applied flip--34.7

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

    4. Applied simplify17.6

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

    6. Applied clear-num17.7

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

    7. Applied simplify8.6

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

    if -1.9881359338705715e-231 < b/2 < 2.104482131095671e+151

    1. Initial program 9.3

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

    if 2.104482131095671e+151 < b/2

    1. Initial program 59.0

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

    2. Taylor expanded around inf 2.4

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

Runtime

Time bar (total: 32.1s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))