Average Error: 33.4 → 8.7
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le -4.291234780139178 \cdot 10^{+129}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le 6.3699601352554695 \cdot 10^{-127}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le 1.6243523938381705 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{a}{b_2} \cdot \left(c \cdot \frac{1}{2}\right) - \left(b_2 + b_2\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (/ c (* (- 1/2) (/ c b_2))) < -4.291234780139178e+129

    1. Initial program 48.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 10.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - 2 \cdot b_2}}{a}\]

    3. Applied simplify2.7

      \[\leadsto \color{blue}{\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2}\]

    if -4.291234780139178e+129 < (/ c (* (- 1/2) (/ c b_2))) < 6.3699601352554695e-127

    1. Initial program 11.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub11.7

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 6.3699601352554695e-127 < (/ c (* (- 1/2) (/ c b_2))) < 1.6243523938381705e+92

    1. Initial program 41.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--41.9

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Applied simplify15.1

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Applied simplify15.1

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied div-inv15.2

      \[\leadsto \color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}}\]

    if 1.6243523938381705e+92 < (/ c (* (- 1/2) (/ c b_2)))

    1. Initial program 57.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--57.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Applied simplify31.0

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Applied simplify31.0

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied div-inv31.0

      \[\leadsto \color{blue}{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}}\]

    8. Taylor expanded around -inf 14.0

      \[\leadsto \frac{c \cdot a}{\color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - b_2\right)} - b_2} \cdot \frac{1}{a}\]

    9. Applied simplify2.6

      \[\leadsto \color{blue}{\frac{c}{\frac{a}{b_2} \cdot \left(c \cdot \frac{1}{2}\right) - \left(b_2 + b_2\right)}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))