Average Error: 25.9 → 2.1
Time: 1.5m
Precision: 64
Internal Precision: 2880
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot a\]
\[\begin{array}{l} \mathbf{if}\;b \le -1.3720909126072433 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{2} \cdot \left(\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b\right)\\ \mathbf{if}\;b \le 7.975363662087551 \cdot 10^{-259}:\\ \;\;\;\;\frac{a}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\\ \mathbf{if}\;b \le 4.180094940320339 \cdot 10^{+92}:\\ \;\;\;\;\frac{a}{2} \cdot \frac{\left(a \cdot c\right) \cdot \left(-4\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{2} \cdot \left(-2 \cdot \frac{a \cdot c}{b}\right)\\ \end{array}\]

Error

Bits error versus b

Bits error versus a

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes
  2. if b < -1.3720909126072433e+154

    1. Initial program 60.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot a\]
    2. Applied simplify60.9

      \[\leadsto \color{blue}{\frac{a}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}\]
    3. Taylor expanded around -inf 0.8

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

    if -1.3720909126072433e+154 < b < 7.975363662087551e-259

    1. Initial program 1.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot a\]
    2. Applied simplify1.6

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

    if 7.975363662087551e-259 < b < 4.180094940320339e+92

    1. Initial program 16.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot a\]
    2. Applied simplify16.0

      \[\leadsto \color{blue}{\frac{a}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}\]
    3. Using strategy rm
    4. Applied flip--16.1

      \[\leadsto \frac{a}{2} \cdot \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}\]
    5. Applied simplify1.9

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

    if 4.180094940320339e+92 < b

    1. Initial program 55.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot a\]
    2. Applied simplify55.3

      \[\leadsto \color{blue}{\frac{a}{2} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}\]
    3. Taylor expanded around inf 3.7

      \[\leadsto \frac{a}{2} \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b}\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (b a c)
  :name "(-b + sqrt(b*b - 4 a c)) / 2a"
  (* (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) 2) a))