Average Error: 19.6 → 6.6
Time: 1.2m
Precision: 64
Internal Precision: 384
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1658649073163418 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c + c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - \frac{b + b}{a + a}\\ \end{array}\\ \mathbf{if}\;b \le 1.2232114636131862 \cdot 10^{+66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c + c}{(\left(\frac{c + c}{b}\right) \cdot a + \left(\left(-b\right) - b\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)}{a + a}} \cdot \sqrt[3]{\frac{\left(2 \cdot \frac{c \cdot a}{b} - b\right) + \left(-b\right)}{a + a}}\right) \cdot \sqrt[3]{\frac{(a \cdot \left(\frac{c}{b}\right) + \left(-b\right))_* + (a \cdot \left(\frac{c}{b}\right) + \left(-b\right))_*}{a + a}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.1658649073163418e+154

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 11.1

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

    3. Applied simplify2.3

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

    if -1.1658649073163418e+154 < b < 1.2232114636131862e+66

    1. Initial program 8.8

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

    3. Applied add-sqr-sqrt8.8

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

    4. Applied sqrt-prod8.9

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

    if 1.2232114636131862e+66 < b

    1. Initial program 26.4

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

    2. Taylor expanded around inf 6.7

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

    3. Applied simplify3.1

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

    4. Taylor expanded around -inf 3.1

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

    6. Applied add-cube-cbrt3.1

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

    8. Applied add-cbrt-cube3.1

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

    9. Applied simplify3.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))