Original	33.0
Target	20.4
Herbie	6.7

Derivation

Split input into 4 regimes
if b < -6.613957471284742e+130
1. Initial program 60.5
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification60.5
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around inf 60.5
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified60.5
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
5. Taylor expanded around -inf 1.8
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
6. Simplified1.8
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
if -6.613957471284742e+130 < b < -9.803705485304467e-285
1. Initial program 33.6
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification33.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around inf 33.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified33.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
5. Using strategy rm
6. Applied flip--33.7
  \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}{2 \cdot a}\]
7. Applied associate-/l/37.9
  \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}}\]
8. Simplified20.9
  \[\leadsto \frac{\color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}\]
9. Using strategy rm
10. Applied times-frac15.8
  \[\leadsto \color{blue}{\frac{a \cdot c}{2 \cdot a} \cdot \frac{4}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\]
11. Simplified8.2
  \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot c\right)} \cdot \frac{4}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]
12. Simplified8.2
  \[\leadsto \left(\frac{1}{2} \cdot c\right) \cdot \color{blue}{\frac{4}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}\]
if -9.803705485304467e-285 < b < 4.353512703078241e+76
1. Initial program 9.6
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification9.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around inf 9.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified9.6
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
if 4.353512703078241e+76 < b
1. Initial program 38.9
  \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Initial simplification38.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]
3. Taylor expanded around inf 38.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
4. Simplified38.9
  \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
5. Using strategy rm
6. Applied flip--60.7
  \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}}{2 \cdot a}\]
7. Applied associate-/l/61.2
  \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}}\]
8. Simplified61.4
  \[\leadsto \frac{\color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right)}\]
9. Taylor expanded around 0 4.6
  \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
10. Simplified4.6
  \[\leadsto \color{blue}{\frac{-b}{a}}\]
Recombined 4 regimes into one program.
Final simplification6.7
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.613957471284742 \cdot 10^{+130}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -9.803705485304467 \cdot 10^{-285}:\\ \;\;\;\;\left(\frac{1}{2} \cdot c\right) \cdot \frac{4}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 4.353512703078241 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Error

Target

Derivation

`if b < -6.613957471284742e+130`

`if -6.613957471284742e+130 < b < -9.803705485304467e-285`

`if -9.803705485304467e-285 < b < 4.353512703078241e+76`

`if 4.353512703078241e+76 < b`

Runtime