Average Error: 33.3 → 10.1
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -9.525999608616756 \cdot 10^{-56}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 2.2573036792948806 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

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

Original33.3
Target19.9
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- b) < -9.525999608616756e-56

    1. Initial program 53.2

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

    2. Taylor expanded around inf 46.5

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

    3. Applied simplify7.8

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

    if -9.525999608616756e-56 < (- b) < 2.2573036792948806e+44

    1. Initial program 14.0

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

    3. Applied clear-num14.1

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

    4. Applied simplify14.1

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

    if 2.2573036792948806e+44 < (- b)

    1. Initial program 35.6

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

    2. Taylor expanded around -inf 6.2

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

    3. Applied simplify6.2

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

  4. Applied simplify10.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -9.525999608616756 \cdot 10^{-56}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 2.2573036792948806 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))