Average Error: 19.6 → 7.9
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.7184540639932095 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\left(\sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \cdot \sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\right) \cdot \sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \le 9.60294828328832 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + {e}^{\left(\log \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)}}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.7184540639932095e+20

    1. Initial program 32.8

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

    2. Taylor expanded around -inf 11.9

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

    3. Applied simplify7.1

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

    5. Applied add-cube-cbrt7.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\left(\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\right) \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{a \cdot 2}\\ \end{array}\]

    if -5.7184540639932095e+20 < b < 9.60294828328832e+82

    1. Initial program 9.5

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

    3. Applied add-exp-log10.9

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

    5. Applied pow110.9

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

    6. Applied log-pow10.9

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

    7. Applied exp-prod11.0

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

    8. Applied simplify11.0

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

    if 9.60294828328832e+82 < b

    1. Initial program 28.1

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

    2. Taylor expanded around inf 6.2

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

    3. Applied simplify2.6

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

  4. Applied simplify7.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -5.7184540639932095 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\left(\sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \cdot \sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\right) \cdot \sqrt[3]{\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \le 9.60294828328832 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + {e}^{\left(\log \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)\right)}}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(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))))