Average Error: 33.0 → 8.4
Time: 31.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.256970379816327 \cdot 10^{+61}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -5.674963253141149 \cdot 10^{-145}:\\ \;\;\;\;\frac{4 \cdot c}{\frac{1}{a} \cdot \left(\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)\right)\right)}\\ \mathbf{elif}\;b \le 2.0296885311959466 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

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

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

Derivation

  1. Split input into 4 regimes

  2. if b < -4.256970379816327e+61

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 56.7

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

    3. Taylor expanded around -inf 3.9

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

    4. Simplified3.9

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

    if -4.256970379816327e+61 < b < -5.674963253141149e-145

    1. Initial program 37.6

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

    3. Applied flip--37.7

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

    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. Simplified21.4

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

    7. Applied associate-/l*13.1

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

    9. Applied div-inv13.2

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

    if -5.674963253141149e-145 < b < 2.0296885311959466e+76

    1. Initial program 11.2

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

    2. Taylor expanded around -inf 11.2

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

    if 2.0296885311959466e+76 < b

    1. Initial program 38.9

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

    3. Applied flip--60.6

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

    4. Applied associate-/l/61.2

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

    5. Simplified61.4

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

    6. Taylor expanded around 0 4.6

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

    7. Simplified4.6

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.256970379816327 \cdot 10^{+61}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -5.674963253141149 \cdot 10^{-145}:\\ \;\;\;\;\frac{4 \cdot c}{\frac{1}{a} \cdot \left(\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)\right)\right)}\\ \mathbf{elif}\;b \le 2.0296885311959466 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Runtime

Time bar (total: 31.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes33.38.45.627.789.9%
herbie shell --seed 2018355 
(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)))