Average Error: 33.3 → 14.4
Time: 27.1s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3045915174446723 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \mathbf{elif}\;b \le 2.2788495809902476 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{-\left(b + \sqrt{b \cdot b + \left(-4 \cdot c\right) \cdot a}\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(b - a \cdot \frac{c}{b}\right)}{2}}{a}\\ \end{array}\]

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

Original33.3
Target20.5
Herbie14.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3045915174446723e-76

    1. Initial program 52.4

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

    2. Initial simplification52.4

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

    3. Taylor expanded around 0 52.4

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

    4. Taylor expanded around -inf 20.6

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

    if -1.3045915174446723e-76 < b < 2.2788495809902476e+126

    1. Initial program 12.4

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

    2. Initial simplification12.4

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

    4. Applied div-inv12.5

      \[\leadsto -\color{blue}{\frac{b + \sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied associate-*l/12.5

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

    7. Simplified12.4

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

    if 2.2788495809902476e+126 < b

    1. Initial program 51.6

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

    2. Initial simplification51.6

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

    3. Taylor expanded around inf 10.6

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

    4. Simplified3.5

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

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3045915174446723 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \mathbf{elif}\;b \le 2.2788495809902476 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{-\left(b + \sqrt{b \cdot b + \left(-4 \cdot c\right) \cdot a}\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(b - a \cdot \frac{c}{b}\right)}{2}}{a}\\ \end{array}\]

Runtime

Time bar (total: 27.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))