Average Error: 32.8 → 9.9
Time: 23.4s
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.858471778140402 \cdot 10^{+95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2136635266576205 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

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

Original32.8
Target20.3
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.858471778140402e+95

    1. Initial program 43.4

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

    2. Initial simplification43.4

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

    4. Applied sub-neg43.4

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

    5. Taylor expanded around -inf 4.0

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

    if -4.858471778140402e+95 < b < 1.2136635266576205e-73

    1. Initial program 12.5

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

    2. Initial simplification12.5

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

    4. Applied *-un-lft-identity12.5

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

    5. Applied associate-/l*12.7

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

    if 1.2136635266576205e-73 < b

    1. Initial program 52.2

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

    2. Initial simplification52.2

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified9.1

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.858471778140402 \cdot 10^{+95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2136635266576205 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 23.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.89.95.326.582.8%
herbie shell --seed 2018340 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))