Average Error: 33.1 → 8.7
Time: 36.4s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.083771773150164 \cdot 10^{+50}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.9267349888076152 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{2 \cdot a}}{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \le 1.8865031203591934 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

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

Original33.1
Target20.6
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.083771773150164e+50

    1. Initial program 56.6

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

    2. Taylor expanded around -inf 4.1

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

    3. Simplified4.1

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

    if -6.083771773150164e+50 < b < -1.9267349888076152e-306

    1. Initial program 29.2

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

    3. Applied flip--29.3

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

    4. Applied associate-/l/33.7

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

    5. Simplified21.8

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

    7. Applied associate-/r*16.6

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

    8. Simplified16.6

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

    if -1.9267349888076152e-306 < b < 1.8865031203591934e+98

    1. Initial program 8.6

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

    2. Taylor expanded around -inf 8.6

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

    if 1.8865031203591934e+98 < b

    1. Initial program 44.1

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

    3. Applied flip--61.6

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

    4. Applied associate-/l/62.1

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

    5. Simplified62.0

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

    6. Taylor expanded around 0 3.9

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

    7. Simplified3.9

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.083771773150164 \cdot 10^{+50}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.9267349888076152 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{2 \cdot a}}{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \le 1.8865031203591934 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 36.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.48.75.625.888%
herbie shell --seed 2018295 
(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)))