Average Error: 33.1 → 11.6
Time: 37.3s
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.338815475246526 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 6.959587268650696 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3313504726342534 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{b} \cdot -2}{2 \cdot a}\\ \end{array}\]

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

Original33.1
Target20.2
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.338815475246526e+154

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -1.338815475246526e+154 < b < 6.959587268650696e-95

    1. Initial program 11.6

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

    if 6.959587268650696e-95 < b < 1.3313504726342534e+154

    1. Initial program 42.3

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

    3. Applied flip-+42.4

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

    4. Applied associate-/l/44.4

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

    5. Simplified14.7

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

    if 1.3313504726342534e+154 < b

    1. Initial program 62.9

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

    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 simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.338815475246526 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 6.959587268650696 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3313504726342534 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{b} \cdot -2}{2 \cdot a}\\ \end{array}\]

Runtime

Time bar (total: 37.3s)Debug logProfile

herbie shell --seed 2018216 
(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)))