Average Error: 33.4 → 8.6
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.5434521988384685 \cdot 10^{+105}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.774986705810536 \cdot 10^{-117}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 9.258556548553494 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

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

Original33.4
Target20.4
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.5434521988384685e+105

    1. Initial program 46.7

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

    2. Taylor expanded around -inf 9.9

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

    3. Applied simplify4.0

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}}\]

    if -4.5434521988384685e+105 < b < 2.774986705810536e-117

    1. Initial program 11.0

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

    if 2.774986705810536e-117 < b < 9.258556548553494e+39

    1. Initial program 38.9

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

    3. Applied flip-+39.0

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

    4. Applied simplify15.9

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

    if 9.258556548553494e+39 < b

    1. Initial program 55.9

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

    2. Taylor expanded around inf 42.4

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

    3. Applied simplify4.2

      \[\leadsto \color{blue}{\left(-1\right) \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify8.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -4.5434521988384685 \cdot 10^{+105}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.774986705810536 \cdot 10^{-117}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 9.258556548553494 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018207 
(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)))