Average Error: 32.9 → 9.5
Time: 57.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.188021985285907 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.1028287840560424 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}{\frac{a \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}\\ \mathbf{elif}\;b \le 2.00427658436863 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot c\right)}{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.9
Target20.3
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.188021985285907e+106

    1. Initial program 46.2

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

    2. Initial simplification46.2

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

    3. Taylor expanded around -inf 3.7

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

    if -4.188021985285907e+106 < b < 2.1028287840560424e-113

    1. Initial program 11.2

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

    2. Initial simplification11.2

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

    4. Applied add-sqr-sqrt11.6

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

    5. Applied associate-/l*11.6

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

    if 2.1028287840560424e-113 < b < 2.00427658436863e+92

    1. Initial program 41.1

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

    2. Initial simplification41.1

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

    4. Applied flip--41.2

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

    5. Applied associate-/l/44.0

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

    6. Simplified18.2

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

    if 2.00427658436863e+92 < b

    1. Initial program 58.2

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

    2. Initial simplification58.2

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

    3. Taylor expanded around inf 2.8

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

    4. Simplified2.8

      \[\leadsto \color{blue}{\frac{-c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.188021985285907 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.1028287840560424 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}{\frac{a \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}\\ \mathbf{elif}\;b \le 2.00427658436863 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot c\right)}{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 57.9s)Debug logProfile

herbie shell --seed 2018277 
(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)))