Average Error: 33.0 → 6.2
Time: 55.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.400267889437694 \cdot 10^{+153}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 5.74603165710677 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 6.040618413936409 \cdot 10^{+135}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

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 < -9.400267889437694e+153

    1. Initial program 60.8

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

    2. Simplified60.8

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

    3. Taylor expanded around -inf 9.8

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

    4. Simplified1.7

      \[\leadsto \frac{\color{blue}{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(-2 \cdot b_2\right))_*}}{a}\]

    if -9.400267889437694e+153 < b_2 < 5.74603165710677e-179

    1. Initial program 9.5

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

    2. Simplified9.5

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

    if 5.74603165710677e-179 < b_2 < 6.040618413936409e+135

    1. Initial program 38.8

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

    2. Simplified38.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied clear-num38.8

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied flip--38.9

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

    7. Applied associate-/r/38.9

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}}\]

    8. Applied add-cube-cbrt38.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}\]

    9. Applied times-frac38.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}} \cdot \frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}\]

    10. Simplified13.6

      \[\leadsto \color{blue}{\frac{0 - a \cdot c}{a}} \cdot \frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]

    11. Simplified13.6

      \[\leadsto \frac{0 - a \cdot c}{a} \cdot \color{blue}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}\]

    12. Taylor expanded around -inf 6.0

      \[\leadsto \color{blue}{\left(-1 \cdot c\right)} \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]

    13. Simplified6.0

      \[\leadsto \color{blue}{\left(-c\right)} \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]
    14. Using strategy rm

    15. Applied distribute-lft-neg-out6.0

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

    16. Simplified5.9

      \[\leadsto -\color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}\]

    if 6.040618413936409e+135 < b_2

    1. Initial program 61.1

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

    2. Simplified61.1

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

    3. Taylor expanded around inf 2.0

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.400267889437694 \cdot 10^{+153}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 5.74603165710677 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 6.040618413936409 \cdot 10^{+135}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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))