Average Error: 20.0 → 7.8
Time: 48.7s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.818264606683512 \cdot 10^{+17}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right) \cdot \left(a \cdot \sqrt[3]{\frac{c}{b}}\right)\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 2.2771361016259362 \cdot 10^{+45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(\frac{c}{b} \cdot a\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right| + \left(-b\right)} \cdot \sqrt{\left|\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right| + \left(-b\right)}}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.818264606683512e+17

    1. Initial program 32.1

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified32.1

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

    5. Applied add-sqr-sqrt32.1

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

    6. Applied rem-sqrt-square32.1

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

    7. Simplified32.1

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

    8. Taylor expanded around 0 7.2

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

    10. Applied add-cube-cbrt7.2

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

    11. Applied associate-*l*7.2

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

    if -2.818264606683512e+17 < b < 2.2771361016259362e+45

    1. Initial program 10.4

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

    3. Applied add-cube-cbrt10.6

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

    4. Applied sqrt-prod10.6

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

    5. Simplified10.6

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

    if 2.2771361016259362e+45 < b

    1. Initial program 25.9

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

    2. Taylor expanded around inf 7.0

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

    3. Simplified3.8

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

    5. Applied add-sqr-sqrt3.8

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

    6. Applied rem-sqrt-square3.8

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

    7. Simplified3.8

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

    9. Applied add-sqr-sqrt3.8

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.818264606683512 \cdot 10^{+17}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right) \cdot \left(a \cdot \sqrt[3]{\frac{c}{b}}\right)\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 2.2771361016259362 \cdot 10^{+45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(\frac{c}{b} \cdot a\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right| + \left(-b\right)} \cdot \sqrt{\left|\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right| + \left(-b\right)}}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 48.7s)Debug logProfile

herbie shell --seed 2018248 +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))))