Average Error: 34.9 → 5.6
Time: 20.5s
Precision: 64
Internal precision: 2688
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;b \le -4.437635090274828 \cdot 10^{+72}:\\
\;\;\;\;\frac{\left(-b\right) + b}{a + a} - \frac{c}{b}\\
\mathbf{if}\;b \le -5.122826773439569 \cdot 10^{-234}:\\
\;\;\;\;\frac{1}{\frac{2}{4 \cdot c} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)\right)}\\
\mathbf{if}\;b \le 5.782486744451723 \cdot 10^{+116}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\end{array}\]
Target
| Original | 34.9 |
|---|
| Comparison | 22.2 |
|---|
| Herbie | 5.6 |
|---|
\[ \begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\end{array} \]
Derivation
- Split input into 4 regimes.
-
if b < -4.437635090274828e+72
Initial program 59.0
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 41.1
\[\leadsto \frac{\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)}{2 \cdot a}\]
Taylor expanded around -inf 41.1
\[\leadsto \frac{\left(-b\right) - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]
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 -4.437635090274828e+72 < b < -5.122826773439569e-234
Initial program 34.5
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm
Applied flip-- 34.6
\[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify 17.7
\[\leadsto \frac{\frac{\color{blue}{a \cdot \left(4 \cdot c\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
- Using strategy
rm
Applied clear-num 17.9
\[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{a \cdot \left(4 \cdot c\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]
Applied simplify 9.2
\[\leadsto \frac{1}{\color{blue}{\frac{2}{4 \cdot c} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)\right)}}\]
if -5.122826773439569e-234 < b < 5.782486744451723e+116
Initial program 9.7
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm
Applied div-sub 9.7
\[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\]
if 5.782486744451723e+116 < b
Initial program 51.1
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 12.8
\[\leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
Taylor expanded around inf 12.8
\[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
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 '#(1065033997 2389885643 4100569014 2620012693 26800780 3144211646)'
(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)))
