Average Error: 33.4 → 10.1
Time: 33.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.8880468196997556 \cdot 10^{-80}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.743299671584071 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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

Original33.4
Target20.6
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.8880468196997556e-80

    1. Initial program 51.8

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

    3. Applied div-inv51.8

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

    4. Taylor expanded around -inf 10.1

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

    5. Simplified10.1

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

    if -2.8880468196997556e-80 < b < 2.743299671584071e+131

    1. Initial program 12.2

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

    2. Taylor expanded around -inf 12.2

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

    4. Applied *-un-lft-identity12.2

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

    5. Applied *-un-lft-identity12.2

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

    6. Applied distribute-lft-out--12.2

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

    7. Applied associate-/l*12.3

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

    if 2.743299671584071e+131 < b

    1. Initial program 53.3

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

    3. Applied div-inv53.3

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

    4. Taylor expanded around inf 2.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.8880468196997556 \cdot 10^{-80}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.743299671584071 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 33.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.110.15.526.682.7%
herbie shell --seed 2018353 
(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)))