Average Error: 32.9 → 8.6
Time: 49.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1144995683522713 \cdot 10^{+98}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.616795426913432 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{\sqrt{a \cdot \left(-4 \cdot c\right) + b \cdot b}} \cdot \sqrt{\sqrt{a \cdot \left(-4 \cdot c\right) + b \cdot b}} - b}\\ \mathbf{elif}\;b \le 6.008237338849469 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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

Original32.9
Target19.9
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 4 regimes

  2. if b < -1.1144995683522713e+98

    1. Initial program 58.4

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

    2. Initial simplification58.4

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

    4. Applied div-sub59.0

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

    5. Taylor expanded around -inf 2.5

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

    6. Simplified2.5

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

    if -1.1144995683522713e+98 < b < -5.616795426913432e-102

    1. Initial program 40.5

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

    2. Initial simplification40.5

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

    4. Applied flip--40.6

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

    5. Applied associate-/l/43.6

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

    6. Simplified16.7

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

    8. Applied associate-/r*12.4

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

    9. Simplified12.4

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

    11. Applied add-sqr-sqrt12.4

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

    12. Applied sqrt-prod12.6

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

    if -5.616795426913432e-102 < b < 6.008237338849469e+153

    1. Initial program 11.4

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

    2. Initial simplification11.4

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

    4. Applied div-sub11.4

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

    if 6.008237338849469e+153 < b

    1. Initial program 60.8

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

    2. Initial simplification60.8

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

    4. Applied div-sub60.8

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

    5. Taylor expanded around inf 2.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1144995683522713 \cdot 10^{+98}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.616795426913432 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{\sqrt{a \cdot \left(-4 \cdot c\right) + b \cdot b}} \cdot \sqrt{\sqrt{a \cdot \left(-4 \cdot c\right) + b \cdot b}} - b}\\ \mathbf{elif}\;b \le 6.008237338849469 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 49.4s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))