Average Error: 33.5 → 9.1
Time: 49.3s
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b/2} \le -4.566008549112679 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b/2} \le 1.1505889895659891 \cdot 10^{-296}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}} - \left(b/2 + b/2\right)}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b/2} \le 4.650924575853574 \cdot 10^{-151}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (/ -1/2 b/2) < -4.566008549112679e+25

    1. Initial program 22.6

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

    3. Applied flip-+22.7

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

    4. Applied simplify16.4

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

    if -4.566008549112679e+25 < (/ -1/2 b/2) < 1.1505889895659891e-296

    1. Initial program 54.2

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

    3. Applied flip-+54.3

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

    4. Applied simplify26.7

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

    5. Taylor expanded around inf 18.2

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

    6. Applied simplify7.6

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

    if 1.1505889895659891e-296 < (/ -1/2 b/2) < 4.650924575853574e-151

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 2.4

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

    if 4.650924575853574e-151 < (/ -1/2 b/2)

    1. Initial program 8.9

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 49.3s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))