Original	33.2
Target	20.5
Herbie	10.4

Derivation

Split input into 4 regimes
if b < -2.2624163142142334e+27 or -2.014594234406352e-62 < b < -1.1320615158913861e-146
1. Initial program 51.7
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification51.7
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around -inf 10.6
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
4. Simplified10.6
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
if -2.2624163142142334e+27 < b < -2.014594234406352e-62
1. Initial program 39.9
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification39.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around 0 39.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified39.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
5. Using strategy rm
6. Applied clear-num39.9
  \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
7. Using strategy rm
8. Applied div-inv39.9
  \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot a\right) \cdot \frac{1}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
9. Applied associate-/r*39.9
  \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
10. Using strategy rm
11. Applied flip--40.0
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\frac{1}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}}\]
12. Applied associate-/r/40.0
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\color{blue}{\frac{1}{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}}\]
13. Applied div-inv40.0
  \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{2 \cdot a}}}{\frac{1}{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}\]
14. Applied times-frac42.9
  \[\leadsto \color{blue}{\frac{1}{\frac{1}{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}} \cdot \frac{\frac{1}{2 \cdot a}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\]
15. Simplified17.8
  \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 4\right)} \cdot \frac{\frac{1}{2 \cdot a}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]
16. Simplified17.8
  \[\leadsto \left(\left(c \cdot a\right) \cdot 4\right) \cdot \color{blue}{\frac{\frac{\frac{1}{2}}{a}}{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} - b}}\]
if -1.1320615158913861e-146 < b < 2.8358752990862995e+119
1. Initial program 11.4
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification11.4
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around 0 11.4
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified11.4
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
if 2.8358752990862995e+119 < b
1. Initial program 49.2
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification49.2
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around 0 49.2
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified49.2
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
5. Taylor expanded around inf 2.7
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
Recombined 4 regimes into one program.
Final simplification10.4
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2624163142142334 \cdot 10^{+27}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.014594234406352 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot 4\right) \cdot \frac{\frac{\frac{1}{2}}{a}}{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le -1.1320615158913861 \cdot 10^{-146}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.8358752990862995 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Target

Derivation

`if b < -2.2624163142142334e+27 or -2.014594234406352e-62 < b < -1.1320615158913861e-146`

`if -2.2624163142142334e+27 < b < -2.014594234406352e-62`

`if -1.1320615158913861e-146 < b < 2.8358752990862995e+119`

`if 2.8358752990862995e+119 < b`

Runtime