Average Error: 34.1 → 5.4
Time: 18.5s
Precision: 64
Internal precision: 2432
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;b \le -1.4229928188391741 \cdot 10^{+144}:\\
\;\;\;\;\frac{\left(-b\right) + b}{a + a} - \frac{c}{b}\\
\mathbf{if}\;b \le 1.912797832812561 \cdot 10^{-298}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c \cdot 4}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + \left(-b\right)}\\
\mathbf{if}\;b \le 2.6629888197799027 \cdot 10^{+103}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\end{array}\]
Target
| Original | 34.1 |
|---|
| Comparison | 21.6 |
|---|
| Herbie | 5.4 |
|---|
\[ \begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\end{array} \]
Derivation
- Split input into 4 regimes.
-
if b < -1.4229928188391741e+144
Initial program 61.6
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 38.7
\[\leadsto \frac{\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)}{2 \cdot a}\]
Taylor expanded around -inf 38.7
\[\leadsto \frac{\left(-b\right) - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]
Applied simplify 0
\[\leadsto \color{blue}{\frac{\left(-b\right) + b}{a + a} - \frac{\frac{c}{b}}{1}}\]
Applied simplify 0
\[\leadsto \frac{\left(-b\right) + b}{a + a} - \color{blue}{\frac{c}{b}}\]
if -1.4229928188391741e+144 < b < 1.912797832812561e-298
Initial program 33.4
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm
Applied flip-- 33.5
\[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify 15.2
\[\leadsto \frac{\frac{\color{blue}{a \cdot \left(4 \cdot c\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
- Using strategy
rm
Applied *-un-lft-identity 15.2
\[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot \left(4 \cdot c\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied times-frac 15.2
\[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\frac{a \cdot \left(4 \cdot c\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
Applied simplify 7.8
\[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c \cdot 4}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + \left(-b\right)}}\]
if 1.912797832812561e-298 < b < 2.6629888197799027e+103
Initial program 9.1
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
if 2.6629888197799027e+103 < b
Initial program 48.2
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 11.4
\[\leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
Taylor expanded around inf 11.4
\[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
Applied simplify 0.0
\[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
Applied simplify 0.0
\[\leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
- Recombined 4 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(FPCore (a b c)
:name "quadm (p42, negative)"
:target
(if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))
(/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))
