Average Error: 33.8 → 6.5
Time: 1.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6100097810064202 \cdot 10^{+95}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -1.8693079465846072 \cdot 10^{-171}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.014307269770461 \cdot 10^{+123}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{2} \cdot \frac{-2}{b}\\ \end{array}\]

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

Original33.8
Target20.9
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.6100097810064202e+95

    1. Initial program 45.2

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

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -2.6100097810064202e+95 < b < -1.8693079465846072e-171

    1. Initial program 6.2

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

    3. Applied div-inv6.4

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

    if -1.8693079465846072e-171 < b < 3.014307269770461e+123

    1. Initial program 30.2

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

    3. Applied flip-+30.5

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

    4. Applied associate-/l/35.6

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

    5. Simplified20.8

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

    7. Applied times-frac15.1

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

    8. Simplified9.8

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

    10. Applied frac-times9.7

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

    if 3.014307269770461e+123 < b

    1. Initial program 60.0

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

    3. Applied flip-+60.0

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

    4. Applied associate-/l/60.1

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

    5. Simplified35.3

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

    7. Applied times-frac34.7

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

    8. Simplified34.1

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

    9. Taylor expanded around inf 2.4

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6100097810064202 \cdot 10^{+95}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -1.8693079465846072 \cdot 10^{-171}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.014307269770461 \cdot 10^{+123}:\\ \;\;\;\;\frac{c \cdot 4}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{2} \cdot \frac{-2}{b}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018215 
(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)))