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
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 46.2
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
3. Simplified46.2
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
4. 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
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 11.2
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
3. Simplified11.2
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
if 7.397988601404757e-150 < b < 2.7301022051416326e+115
1. Initial program 38.7
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 38.6
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
3. Simplified38.7
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
4. Using strategy rm
5. Applied clear-num38.7
  \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
6. Using strategy rm
7. Applied div-inv38.7
  \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot a\right) \cdot \frac{1}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
8. Applied associate-/r*38.7
  \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}\]
9. Simplified38.7
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\color{blue}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]
10. Using strategy rm
11. Applied flip--38.8
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\frac{1}{\color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}}}\]
12. Applied associate-/r/38.8
  \[\leadsto \frac{\frac{1}{2 \cdot a}}{\color{blue}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}}\]
13. Applied add-cube-cbrt39.0
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\]
14. Applied times-frac39.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}} \cdot \frac{\sqrt[3]{\frac{1}{2 \cdot a}}}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}\]
15. Simplified14.8
  \[\leadsto \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2 \cdot a}}}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}\]
16. Simplified14.8
  \[\leadsto \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + b}}\]
if 2.7301022051416326e+115 < b
1. Initial program 59.3
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 59.3
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
3. Simplified59.4
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
4. Taylor expanded around inf 1.9
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
5. Simplified1.9
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
Recombined 4 regimes into one program.
Final simplification9.0
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3801296442484604 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 7.397988601404757 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.7301022051416326 \cdot 10^{+115}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(\left(c \cdot -4\right) \cdot a\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

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