Error

Target

Derivation

1. Split input into 3 regimes

2. if b < -1.2596647331824642e-31

   1. Initial program 53.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified53.7

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity53.7

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{\color{blue}{1 \cdot (\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   5. Applied sqrt-prod53.7

      \[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{\sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   6. Applied add-cube-cbrt57.2

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - \sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]

   7. Applied prod-diff58.1

      \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}}{2}}{a}\]

   8. Simplified53.8

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}{2}}{a}\]

   9. Simplified53.7

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt62.9

       \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) + 0}{2}}{a}\]

   12. Applied neg-mul-162.9

       \[\leadsto \frac{\frac{\left(\color{blue}{-1 \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} - \sqrt{b} \cdot \sqrt{b}\right) + 0}{2}}{a}\]

   13. Applied prod-diff62.9

       \[\leadsto \frac{\frac{\color{blue}{\left((-1 \cdot \left(\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \left(-\sqrt{b} \cdot \sqrt{b}\right))_* + (\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(\sqrt{b} \cdot \sqrt{b}\right))_*\right)} + 0}{2}}{a}\]

   14. Simplified62.9

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\left(-b\right) - \sqrt{(b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right))_*}\right)} + (\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(\sqrt{b} \cdot \sqrt{b}\right))_*\right) + 0}{2}}{a}\]

   15. Simplified53.7

       \[\leadsto \frac{\frac{\left(\left(\left(-b\right) - \sqrt{(b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right))_*}\right) + \color{blue}{\left(\left(-b\right) + b\right)}\right) + 0}{2}}{a}\]

   16. Taylor expanded around -inf 6.7

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   17. Simplified6.7

       \[\leadsto \color{blue}{-\frac{c}{b}}\]

   if -1.2596647331824642e-31 < b < 1.3977277462270081e+122

   1. Initial program 13.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified13.5

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity13.5

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{\color{blue}{1 \cdot (\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   5. Applied sqrt-prod13.5

      \[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{\sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   6. Applied add-cube-cbrt13.7

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - \sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]

   7. Applied prod-diff13.8

      \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}}{2}}{a}\]

   8. Simplified13.6

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}{2}}{a}\]

   9. Simplified13.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt13.5

       \[\leadsto \frac{\frac{\left(\left(-\sqrt{\color{blue}{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}\right) - b\right) + 0}{2}}{a}\]

   12. Applied sqrt-prod13.7

       \[\leadsto \frac{\frac{\left(\left(-\color{blue}{\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}\right) - b\right) + 0}{2}}{a}\]

   13. Applied distribute-lft-neg-in13.7

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(-\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}} - b\right) + 0}{2}}{a}\]

   14. Applied fma-neg13.7

       \[\leadsto \frac{\frac{\color{blue}{(\left(-\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*} + 0}{2}}{a}\]

   if 1.3977277462270081e+122 < b

   1. Initial program 50.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified50.6

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity50.6

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{\color{blue}{1 \cdot (\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   5. Applied sqrt-prod50.6

      \[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{\sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]

   6. Applied add-cube-cbrt50.7

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - \sqrt{1} \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]

   7. Applied prod-diff51.7

      \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}}{2}}{a}\]

   8. Simplified51.6

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot \left(\sqrt{1}\right) + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt{1}\right))_*}{2}}{a}\]

   9. Simplified50.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt50.5

       \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) + 0}{2}}{a}\]

   12. Applied neg-mul-150.5

       \[\leadsto \frac{\frac{\left(\color{blue}{-1 \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} - \sqrt{b} \cdot \sqrt{b}\right) + 0}{2}}{a}\]

   13. Applied prod-diff50.5

       \[\leadsto \frac{\frac{\color{blue}{\left((-1 \cdot \left(\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \left(-\sqrt{b} \cdot \sqrt{b}\right))_* + (\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(\sqrt{b} \cdot \sqrt{b}\right))_*\right)} + 0}{2}}{a}\]

   14. Simplified50.6

       \[\leadsto \frac{\frac{\left(\color{blue}{\left(\left(-b\right) - \sqrt{(b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right))_*}\right)} + (\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(\sqrt{b} \cdot \sqrt{b}\right))_*\right) + 0}{2}}{a}\]

   15. Simplified50.6

       \[\leadsto \frac{\frac{\left(\left(\left(-b\right) - \sqrt{(b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right))_*}\right) + \color{blue}{\left(\left(-b\right) + b\right)}\right) + 0}{2}}{a}\]

   16. Taylor expanded around inf 3.7

       \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2596647331824642 \cdot 10^{-31}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.3977277462270081 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{(\left(-\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))