Average Error: 33.9 → 10.3
Time: 51.2s
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -1.6867480356295008 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot a\right) \cdot \frac{c}{b/2} - \left(b/2 + b/2\right)}{a}\\ \mathbf{if}\;b/2 \le 3.2638354487258445 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}} - \left(b/2 + b/2\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b/2 < -1.6867480356295008e+140

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 10.4

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

    3. Applied simplify2.1

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

    if -1.6867480356295008e+140 < b/2 < 3.2638354487258445e-106

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 3.2638354487258445e-106 < b/2

    1. Initial program 50.8

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

    3. Applied flip-+50.9

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

    4. Applied simplify24.4

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

    5. Taylor expanded around inf 22.1

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

    6. Applied simplify10.4

      \[\leadsto \color{blue}{\frac{c}{\frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}} - \left(b/2 + b/2\right)}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 51.2s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))