Average Error: 32.7 → 9.1
Time: 39.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5896964098726769 \cdot 10^{+44}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -4.0932931157804415 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b\right)}\\ \mathbf{elif}\;b \le 5.147430478402213 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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

Original32.7
Target20.2
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.5896964098726769e+44

    1. Initial program 55.8

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

    2. Initial simplification55.8

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

    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 -1.5896964098726769e+44 < b < -4.0932931157804415e-120

    1. Initial program 37.1

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

    2. Initial simplification37.1

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

    4. Applied div-inv37.2

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

    6. Applied flip--37.2

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

    7. Applied frac-times40.8

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

    8. Simplified19.8

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

    9. Simplified19.8

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

    if -4.0932931157804415e-120 < b < 5.147430478402213e+125

    1. Initial program 11.3

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

    2. Initial simplification11.3

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

    3. Taylor expanded around 0 11.3

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

    if 5.147430478402213e+125 < b

    1. Initial program 50.7

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

    2. Initial simplification50.7

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

    4. Applied div-inv50.7

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

    5. Taylor expanded around inf 2.3

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5896964098726769 \cdot 10^{+44}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -4.0932931157804415 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b\right)}\\ \mathbf{elif}\;b \le 5.147430478402213 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 39.4s)Debug logProfile

herbie shell --seed 2018278 
(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)))