Average Error: 33.4 → 8.9
Time: 1.4m
Precision: 64
Internal Precision: 8192
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -1.0425438267508959 \cdot 10^{-10}:\\ \;\;\;\;\frac{c}{\frac{b/2}{\frac{-1}{2}}}\\ \mathbf{if}\;b/2 \le -4.9359430884765427 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}{a}\\ \mathbf{if}\;b/2 \le 2.1163211128201421 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}} - \left(b/2 + b/2\right)}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b/2 < -1.0425438267508959e-10

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 17.2

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

    3. Applied simplify6.1

      \[\leadsto \color{blue}{\frac{c}{\frac{b/2}{\frac{-1}{2}}}}\]

    if -1.0425438267508959e-10 < b/2 < -4.9359430884765427e-250

    1. Initial program 27.2

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

    3. Applied flip--27.4

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

    4. Applied simplify17.6

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

    5. Applied simplify17.6

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}}{a}\]

    if -4.9359430884765427e-250 < b/2 < 2.1163211128201421e+74

    1. Initial program 9.6

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

    3. Applied clear-num9.7

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

    if 2.1163211128201421e+74 < b/2

    1. Initial program 39.2

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

    2. Taylor expanded around inf 9.8

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

    3. Applied simplify5.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))