Average Error: 19.8 → 6.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(c \cdot 4\right) \cdot \left(-a\right)}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + b}}\\ \end{array} \le -7.362640111226769 \cdot 10^{+282}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2}{\frac{b}{a}} - \frac{b + b}{c}}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(c \cdot 4\right) \cdot \left(-a\right)}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + b}}\\ \end{array} \le 8.238263215483767 \cdot 10^{+276}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2}{\frac{b}{a}} - \frac{b + b}{c}}\\ \end{array}\]

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 2 regimes

  2. if (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* a c) 4)))) (* 2 a)) (/ (* 2 c) (/ (* (* c 4) (- a)) (+ (sqrt (- (* b b) (* (* a c) 4))) b)))) < -7.362640111226769e+282 or 8.238263215483767e+276 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* a c) 4)))) (* 2 a)) (/ (* 2 c) (/ (* (* c 4) (- a)) (+ (sqrt (- (* b b) (* (* a c) 4))) b))))

    1. Initial program 44.7

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

    2. Applied simplify44.6

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

    3. Taylor expanded around inf 31.1

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

    4. Applied simplify28.7

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

    5. Taylor expanded around -inf 14.1

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

    6. Applied simplify12.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2}{\frac{b}{a}} - \frac{b + b}{c}}\\ \end{array}}\]

    if -7.362640111226769e+282 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* a c) 4)))) (* 2 a)) (/ (* 2 c) (/ (* (* c 4) (- a)) (+ (sqrt (- (* b b) (* (* a c) 4))) b)))) < 8.238263215483767e+276

    1. Initial program 1.5

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

    2. Applied simplify1.5

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

    4. Applied add-cube-cbrt1.9

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2019053 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))