Average Error: 19.8 → 13.8
Time: 31.4s
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[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \left(\sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le -2.9857892093808253 \cdot 10^{-202}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \left(\sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le 5.5827589937171 \cdot 10^{-318}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}}\right)\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\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 2 regimes

  2. if (if (>= b 0) (/ (* 2 c) (- (- b) (* (* (cbrt (sqrt (fma (* 4 a) (- c) (* b b)))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))))) (/ (- (sqrt (fma (* 4 a) (- c) (* b b))) b) (* a 2))) < -2.9857892093808253e-202 or 5.5827589937171e-318 < (if (>= b 0) (/ (* 2 c) (- (- b) (* (* (cbrt (sqrt (fma (* 4 a) (- c) (* b b)))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))))) (/ (- (sqrt (fma (* 4 a) (- c) (* b b))) b) (* a 2)))

    1. Initial program 14.5

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

    2. Initial simplification14.5

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

    4. Applied add-sqr-sqrt14.5

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

    5. Applied sqrt-prod14.6

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

    if -2.9857892093808253e-202 < (if (>= b 0) (/ (* 2 c) (- (- b) (* (* (cbrt (sqrt (fma (* 4 a) (- c) (* b b)))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))) (cbrt (sqrt (fma (* 4 a) (- c) (* b b))))))) (/ (- (sqrt (fma (* 4 a) (- c) (* b b))) b) (* a 2))) < 5.5827589937171e-318

    1. Initial program 34.6

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

    2. Initial simplification34.6

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

    3. Taylor expanded around 0 10.3

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

    5. Applied add-log-exp11.6

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \left(\sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le -2.9857892093808253 \cdot 10^{-202}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \left(\sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le 5.5827589937171 \cdot 10^{-318}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}}\right)\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 31.4s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(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))))