Average Error: 33.4 → 8.6
Time: 43.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.9292673416568248:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.9381783838962572 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{2 \cdot a}}{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \le -5.166906115776919 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{a} \cdot \frac{a}{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}\\ \mathbf{elif}\;b \le 7.791433825346608 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(\frac{1}{2} \cdot \frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]

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

Original33.4
Target20.6
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if b < -0.9292673416568248

    1. Initial program 55.5

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

    3. Applied div-sub56.2

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

    4. Taylor expanded around -inf 5.8

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

    5. Simplified5.8

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

    if -0.9292673416568248 < b < -1.9381783838962572e-66

    1. Initial program 37.2

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

    3. Applied flip--37.3

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

    4. Applied associate-/l/40.8

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

    5. Simplified18.8

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

    7. Applied associate-/r*14.4

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

    8. Simplified14.4

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

    if -1.9381783838962572e-66 < b < -5.166906115776919e-157

    1. Initial program 27.2

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

    3. Applied flip--27.4

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

    4. Applied associate-/l/31.9

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

    5. Simplified24.1

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

    7. Applied times-frac16.8

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

    8. Simplified16.8

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

    9. Simplified16.8

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

    if -5.166906115776919e-157 < b < 7.791433825346608e+141

    1. Initial program 10.4

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

    3. Applied div-sub10.4

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

    if 7.791433825346608e+141 < b

    1. Initial program 56.3

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

    3. Applied div-sub56.3

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

    4. Taylor expanded around inf 2.6

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{\left(\frac{1}{2} \cdot \frac{b}{a} - \frac{c}{b}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.9292673416568248:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.9381783838962572 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{2 \cdot a}}{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \le -5.166906115776919 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{a} \cdot \frac{a}{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}\\ \mathbf{elif}\;b \le 7.791433825346608 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(\frac{1}{2} \cdot \frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]

Runtime

Time bar (total: 43.0s)Debug logProfile

herbie shell --seed 2018346 
(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)))