Error

Target

Derivation

1. Split input into 4 regimes

2. if (- b) < -9.670079395517677e+115

   1. Initial program 48.0

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

   2. Taylor expanded around inf 3.3

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

   3. Simplified3.3

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

   if -9.670079395517677e+115 < (- b) < 1.094348108854397e-160

   1. Initial program 10.4

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

   3. Applied add-cube-cbrt11.2

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

   4. Applied add-cube-cbrt11.3

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

   5. Applied prod-diff11.4

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

   6. Simplified10.4

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

   7. Simplified10.3

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

   if 1.094348108854397e-160 < (- b) < 1.1483194775251371e+132

   1. Initial program 39.9

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

   3. Applied flip--40.0

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

   4. Applied associate-/l/43.0

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

   5. Simplified18.6

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

   7. Applied associate-/r*13.6

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

   8. Simplified13.6

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

   10. Applied *-un-lft-identity13.6

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

   11. Applied add-cube-cbrt14.5

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

   12. Applied times-frac14.5

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

   13. Simplified14.4

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

   14. Simplified6.3

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

   if 1.1483194775251371e+132 < (- b)

   1. Initial program 60.9

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

   2. Taylor expanded around -inf 14.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a}\]
3. Recombined 4 regimes into one program.

4. Final simplification9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-b \le -9.670079395517677 \cdot 10^{+115}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;-b \le 1.094348108854397 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{elif}\;-b \le 1.1483194775251371 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{4 \cdot c}{2}}}{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b} \cdot \left(\sqrt[3]{\frac{4}{\frac{2}{c}}} \cdot \sqrt[3]{\frac{4}{\frac{2}{c}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -2}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018216 +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)))