Original	32.9
Target	20.6
Herbie	9.0

Derivation

Split input into 4 regimes
if b < -1.3801296442484604e+103
1. Initial program 46.2
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification46.2
  \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\]
3. Taylor expanded around -inf 4.4
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
if -1.3801296442484604e+103 < b < 7.397988601404757e-150
1. Initial program 11.2
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification11.2
  \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\]
3. Using strategy rm
4. Applied sub-neg11.2
  \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a}\]
if 7.397988601404757e-150 < b < 2.7301022051416326e+115
1. Initial program 38.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification38.7
  \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\]
3. Using strategy rm
4. Applied sub-neg38.7
  \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a}\]
5. Using strategy rm
6. Applied *-un-lft-identity38.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b\right)}}{2 \cdot a}\]
7. Applied associate-/l*38.7
  \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}}\]
8. Using strategy rm
9. Applied div-inv38.7
  \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot a\right) \cdot \frac{1}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}}\]
10. Applied associate-/r*38.7
  \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{\frac{1}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}}\]
11. Simplified38.7
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\color{blue}{\frac{1}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}}\]
12. Using strategy rm
13. Applied flip--38.8
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\frac{1}{\color{blue}{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot b}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} + b}}}}\]
14. Applied associate-/r/38.8
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\color{blue}{\frac{1}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot b} \cdot \left(\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} + b\right)}}\]
15. Applied add-cube-cbrt39.0
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}}{\frac{1}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot b} \cdot \left(\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} + b\right)}\]
16. Applied times-frac39.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{1}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot b}} \cdot \frac{\sqrt[3]{\frac{1}{2 \cdot a}}}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} + b}}\]
17. Simplified14.8
  \[\leadsto \color{blue}{\left(\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2 \cdot a}}}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} + b}\]
18. Simplified14.8
  \[\leadsto \left(\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} + b}}\]
if 2.7301022051416326e+115 < b
1. Initial program 59.3
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification59.3
  \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\]
3. Using strategy rm
4. Applied sub-neg59.3
  \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a}\]
5. Taylor expanded around inf 1.9
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
6. Simplified1.9
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
Recombined 4 regimes into one program.
Final simplification9.0
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3801296442484604 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 7.397988601404757 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.7301022051416326 \cdot 10^{+115}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Try it out

Target

Derivation

`if b < -1.3801296442484604e+103`

`if -1.3801296442484604e+103 < b < 7.397988601404757e-150`

`if 7.397988601404757e-150 < b < 2.7301022051416326e+115`

`if 2.7301022051416326e+115 < b`

Runtime

In
Out