Average Error: 33.7 → 9.4
Time: 54.6s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3903436493377258 \cdot 10^{+56}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.257503605777414 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*} - b} \cdot \left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;b \le 2.039246018158061 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.7
Herbie9.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3903436493377258e+56

    1. Initial program 56.8

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

    2. Initial simplification56.8

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

    3. Taylor expanded around -inf 3.3

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

    4. Simplified3.3

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

    if -1.3903436493377258e+56 < b < -5.257503605777414e-68

    1. Initial program 42.3

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

    2. Initial simplification42.4

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

    4. Applied clear-num42.4

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

    6. Applied div-inv42.5

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

    7. Applied associate-/r*42.5

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

    9. Applied flip--42.6

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

    10. Applied associate-/r/42.6

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

    11. Applied add-cube-cbrt42.8

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

    12. Applied times-frac42.8

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

    13. Simplified15.9

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(0 - \left(a \cdot c\right) \cdot -4\right)\right)} \cdot \frac{\sqrt[3]{\frac{1}{2 \cdot a}}}{\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}\]

    14. Simplified15.9

      \[\leadsto \left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(0 - \left(a \cdot c\right) \cdot -4\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*} - b}}\]

    if -5.257503605777414e-68 < b < 2.039246018158061e+126

    1. Initial program 13.0

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

    2. Initial simplification12.9

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

    4. Applied clear-num13.1

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

    6. Applied div-inv13.1

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

    7. Applied associate-/r*13.1

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

    if 2.039246018158061e+126 < b

    1. Initial program 52.3

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

    2. Initial simplification52.3

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

    3. Taylor expanded around inf 3.2

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3903436493377258 \cdot 10^{+56}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.257503605777414 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{1}{2}}{a}}}{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*} - b} \cdot \left(\left(\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;b \le 2.039246018158061 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 54.6s)Debug logProfile

herbie shell --seed 2018255 +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)))