Average Error: 32.8 → 9.1
Time: 1.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.4810568855201823 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{if}\;b_2 \le 4.0760169085997324 \cdot 10^{-125}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \left(\sqrt[3]{-b_2}\right) + \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}{a}\\ \mathbf{if}\;b_2 \le 9.28447240618529 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

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 < -3.4810568855201823e+100

    1. Initial program 44.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 4.2

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]

    if -3.4810568855201823e+100 < b_2 < 4.0760169085997324e-125

    1. Initial program 10.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt11.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-def11.1

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \left(\sqrt[3]{-b_2}\right) + \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}}{a}\]

    if 4.0760169085997324e-125 < b_2 < 9.28447240618529e+96

    1. Initial program 39.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+40.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Applied simplify16.5

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    if 9.28447240618529e+96 < b_2

    1. Initial program 58.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 13.5

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b_2}}}{a}\]

    3. Applied simplify2.9

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +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))