Average Error: 33.0 → 12.7
Time: 32.8s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.9960999297428996 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -2.1110849490305294 \cdot 10^{-131}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 2.0766812404400347 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{1}{2}\right) \cdot \frac{c}{b_2} - \left(b_2 + b_2\right)}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.9960999297428996e+90

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 14.6

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

    if -2.9960999297428996e+90 < b_2 < -2.1110849490305294e-131

    1. Initial program 39.1

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

    3. Applied flip--39.2

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

    4. Applied associate-/l/42.1

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

    5. Simplified18.3

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

    if -2.1110849490305294e-131 < b_2 < 2.0766812404400347e+50

    1. Initial program 11.9

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

    3. Applied clear-num12.0

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

    if 2.0766812404400347e+50 < b_2

    1. Initial program 35.4

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

    3. Applied add-cube-cbrt35.7

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

    4. Applied fma-neg35.7

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

    5. Taylor expanded around inf 62.4

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

    6. Simplified6.1

      \[\leadsto \frac{\color{blue}{\frac{c}{b_2} \cdot \left(\frac{1}{2} \cdot a\right) - \left(b_2 + b_2\right)}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.9960999297428996 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -2.1110849490305294 \cdot 10^{-131}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 2.0766812404400347 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{1}{2}\right) \cdot \frac{c}{b_2} - \left(b_2 + b_2\right)}{a}\\ \end{array}\]

Runtime

Time bar (total: 32.8s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))