Average Error: 33.7 → 11.6
Time: 7.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.15801717818395 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.679632723565992 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + \left(a \cdot c\right) \cdot 4}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.383546136172223 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{b}{a} \cdot \frac{1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.7
Target21.1
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.15801717818395e+153

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 14.5

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

    if -4.15801717818395e+153 < b < -5.679632723565992e-68

    1. Initial program 43.6

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

    3. Applied div-inv43.6

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

    5. Applied flip--43.6

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

    6. Applied associate-*l/43.7

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

    7. Simplified11.2

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

    if -5.679632723565992e-68 < b < 2.383546136172223e+95

    1. Initial program 13.4

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

    3. Applied div-inv13.5

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

    if 2.383546136172223e+95 < b

    1. Initial program 44.1

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

    3. Applied div-sub44.1

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

    4. Taylor expanded around inf 3.9

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{\frac{1}{2} \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.15801717818395 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.679632723565992 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + \left(a \cdot c\right) \cdot 4}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.383546136172223 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{b}{a} \cdot \frac{1}{2}\\ \end{array}\]

Runtime

Time bar (total: 7.2m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))