Average Error: 34.0 → 5.9
Time: 1.0m
Precision: 64
Internal precision: 2432
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;b \le -5.224339320607286 \cdot 10^{+142}:\\
\;\;\;\;\frac{\left(-b\right) + b}{a + a} - \frac{c}{b}\\
\mathbf{if}\;b \le -6.9528875814338765 \cdot 10^{-292}:\\
\;\;\;\;\frac{1}{\frac{2}{c \cdot 4} \cdot \left(\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + \left(-b\right)\right)}\\
\mathbf{if}\;b \le 2.4225703873767033 \cdot 10^{+115}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\
\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\end{array}\]
Target
| Original | 34.0 |
|---|
| Comparison | 21.5 |
|---|
| Herbie | 5.9 |
|---|
\[ \begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\end{array} \]
Derivation
- Split input into 4 regimes.
-
if b < -5.224339320607286e+142
Initial program 62.1
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 40.3
\[\leadsto \frac{\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)}{2 \cdot a}\]
Taylor expanded around -inf 40.3
\[\leadsto \frac{\left(-b\right) - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]
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 -5.224339320607286e+142 < b < -6.9528875814338765e-292
Initial program 35.1
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm
Applied flip-- 35.2
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify 16.2
\[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
- Using strategy
rm
Applied clear-num 16.3
\[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]
Applied simplify 8.5
\[\leadsto \frac{1}{\color{blue}{\frac{2}{c \cdot 4} \cdot \left(\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + \left(-b\right)\right)}}\]
if -6.9528875814338765e-292 < b < 2.4225703873767033e+115
Initial program 9.1
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm
Applied clear-num 9.3
\[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]
if 2.4225703873767033e+115 < b
Initial program 50.3
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Applied taylor 12.4
\[\leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
Taylor expanded around inf 12.4
\[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
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 '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (a b c)
:name "quadm (p42, negative)"
: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)))
