Average Error: 34.0 → 7.1
Time: 5.0s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.887155650852303 \cdot 10^{94}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \mathbf{elif}\;b_2 \le -2.0531844483893699 \cdot 10^{-250}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.15077727037589584 \cdot 10^{109}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

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 < -1.887155650852303e94

    1. Initial program 45.7

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

    2. Simplified45.7

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

    4. Applied div-inv45.8

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

    5. Taylor expanded around -inf 3.9

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

    6. Simplified3.9

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

    if -1.887155650852303e94 < b_2 < -2.0531844483893699e-250

    1. Initial program 9.2

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

    2. Simplified9.2

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

    4. Applied div-inv9.4

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

    if -2.0531844483893699e-250 < b_2 < 3.15077727037589584e109

    1. Initial program 30.6

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

    2. Simplified30.6

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

    4. Applied div-inv30.6

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

    6. Applied flip--30.6

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

    7. Applied associate-*l/30.7

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

    8. Simplified15.7

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

    10. Applied distribute-rgt-neg-out15.7

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

    11. Applied distribute-frac-neg15.7

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

    12. Simplified10.0

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

    if 3.15077727037589584e109 < b_2

    1. Initial program 59.7

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

    2. Simplified59.7

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

    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 simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.887155650852303 \cdot 10^{94}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \mathbf{elif}\;b_2 \le -2.0531844483893699 \cdot 10^{-250}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.15077727037589584 \cdot 10^{109}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))