Average Error: 19.3 → 6.4
Time: 35.7s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.623284355737863 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot (c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot (c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 8.321168663371558 \cdot 10^{+130}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\frac{2 \cdot c}{(c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{(e^{\log_* (1 + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c})} - 1)^*}}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.623284355737863e+149

    1. Initial program 59.0

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

    2. Taylor expanded around inf 59.0

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

    3. Simplified59.0

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

    4. Taylor expanded around -inf 11.0

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

    5. Simplified2.2

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

    if -2.623284355737863e+149 < b < 8.321168663371558e+130

    1. Initial program 8.4

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

    2. Initial simplification8.4

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

    if 8.321168663371558e+130 < b

    1. Initial program 34.1

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

    2. Taylor expanded around inf 7.0

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

    3. Simplified2.3

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

    5. Applied associate-/r*2.3

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

    7. Applied add-sqr-sqrt2.3

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

    9. Applied expm1-log1p-u2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\frac{2 \cdot c}{(c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{(e^{\log_* (1 + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c})} - 1)^*} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.623284355737863 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot (c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot (c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 8.321168663371558 \cdot 10^{+130}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\frac{2 \cdot c}{(c \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{(e^{\log_* (1 + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c})} - 1)^*}}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 35.7s)Debug logProfile

herbie shell --seed 2018234 +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))))