Average Error: 33.2 → 7.5
Time: 2.3m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3676157084219913 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b + b\right)}\\ \mathbf{if}\;b \le -9.377661527054922 \cdot 10^{-288}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \mathbf{if}\;b \le 7.574837362105014 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

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

Original33.2
Target20.5
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3676157084219913e+111

    1. Initial program 59.1

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

    3. Applied flip--59.1

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

    4. Applied simplify33.3

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

    5. Applied simplify33.3

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

    7. Applied *-un-lft-identity33.3

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

    8. Applied times-frac33.3

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

    9. Applied simplify31.3

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

    10. Taylor expanded around -inf 7.3

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

    11. Applied simplify2.9

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

    if -2.3676157084219913e+111 < b < -9.377661527054922e-288

    1. Initial program 32.9

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

    3. Applied flip--33.0

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

    4. Applied simplify14.4

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

    5. Applied simplify14.4

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

    7. Applied *-un-lft-identity14.4

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

    8. Applied times-frac14.4

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

    9. Applied simplify7.4

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

    if -9.377661527054922e-288 < b < 7.574837362105014e-10

    1. Initial program 11.5

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

    3. Applied div-inv11.6

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

    if 7.574837362105014e-10 < b

    1. Initial program 29.4

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

    2. Taylor expanded around inf 8.3

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

    3. Applied simplify8.3

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018170 
(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)))