Average Error: 19.3 → 7.0
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le -1.0439296684993796 \cdot 10^{+241}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le -1.9180879887997911 \cdot 10^{-295}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le 2.3689218286716864 \cdot 10^{-293}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{a}{b}}} \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right) \cdot \left(2 \cdot c\right)\right) \cdot \left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le 1.1619586500950238 \cdot 10^{+284}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))) < -1.0439296684993796e+241 or 1.1619586500950238e+284 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

    1. Initial program 55.1

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

    2. Applied simplify55.1

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around -inf 22.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    4. Applied simplify16.4

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 1 - \frac{b}{a}\\ \end{array}}\]

    if -1.0439296684993796e+241 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))) < -1.9180879887997911e-295 or 2.3689218286716864e-293 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))) < 1.1619586500950238e+284

    1. Initial program 2.7

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

    2. Applied simplify2.7

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt3.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\]

    if -1.9180879887997911e-295 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))) < 2.3689218286716864e-293

    1. Initial program 35.6

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

    2. Applied simplify35.6

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 12.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\]

    4. Applied simplify9.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{a}{b} \cdot \left(2 \cdot c\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \sqrt[3]{\frac{a}{b}}\right)} \cdot \left(2 \cdot c\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}\]

    7. Applied associate-*l*9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \left(\sqrt[3]{\frac{a}{b}} \cdot \left(2 \cdot c\right)\right)} - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{a}{b}}} \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right)} \cdot \left(2 \cdot c\right)\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify7.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le -1.0439296684993796 \cdot 10^{+241}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le -1.9180879887997911 \cdot 10^{-295}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le 2.3689218286716864 \cdot 10^{-293}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{a}{b}}} \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{a}{b}}}\right) \cdot \left(2 \cdot c\right)\right) \cdot \left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \le 1.1619586500950238 \cdot 10^{+284}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2019053 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))