Average Error: 33.1 → 12.0
Time: 1.2m
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.386154120814377 \cdot 10^{+81}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_* \cdot 2}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.620267927542324 \cdot 10^{-92}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right))_*}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.495342654853265 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \left(c \cdot 4\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.2
Herbie12.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.386154120814377e+81

    1. Initial program 41.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 10.9

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\]

    3. Simplified5.0

      \[\leadsto \frac{\color{blue}{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_* \cdot 2}}{2 \cdot a}\]

    if -1.386154120814377e+81 < b < 6.620267927542324e-92

    1. Initial program 12.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt12.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    4. Applied fma-def12.8

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right))_*}}{2 \cdot a}\]

    if 6.620267927542324e-92 < b < 6.495342654853265e+153

    1. Initial program 42.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+42.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Applied associate-/l/44.5

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]

    5. Simplified14.7

      \[\leadsto \frac{\color{blue}{\left(c \cdot 4\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    if 6.495342654853265e+153 < b

    1. Initial program 62.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 13.6

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.386154120814377 \cdot 10^{+81}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_* \cdot 2}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.620267927542324 \cdot 10^{-92}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right))_*}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.495342654853265 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot \left(c \cdot 4\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))