Average Error: 33.2 → 8.4
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.498957602494875 \cdot 10^{+97}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.5297967170042262 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 4\right) \cdot c}{a \cdot 2}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)}\\ \mathbf{elif}\;b \le 1.2969338258968005 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot (a \cdot \left(\frac{c}{b}\right) + \left(-b\right))_*}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.5
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.498957602494875e+97

    1. Initial program 58.5

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

    3. Applied div-inv58.5

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]

    4. Taylor expanded around -inf 2.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    5. Simplified2.6

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

    if -7.498957602494875e+97 < b < -1.5297967170042262e-198

    1. Initial program 35.6

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

    3. Applied div-inv35.7

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]
    4. Using strategy rm

    5. Applied flip--35.7

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{1}{2 \cdot a}\]

    6. Applied associate-*l/35.8

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Simplified14.3

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

    if -1.5297967170042262e-198 < b < 1.2969338258968005e+123

    1. Initial program 9.9

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

    3. Applied div-inv10.0

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 1.2969338258968005e+123 < b

    1. Initial program 49.8

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

    2. Taylor expanded around inf 11.4

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

    3. Simplified3.5

      \[\leadsto \frac{\color{blue}{2 \cdot (a \cdot \left(\frac{c}{b}\right) + \left(-b\right))_*}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.498957602494875 \cdot 10^{+97}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.5297967170042262 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 4\right) \cdot c}{a \cdot 2}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)}\\ \mathbf{elif}\;b \le 1.2969338258968005 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot (a \cdot \left(\frac{c}{b}\right) + \left(-b\right))_*}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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