Average Error: 33.7 → 8.6
Time: 2.0m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l}
\mathbf{if}\;b \le -1.3907652222998155 \cdot 10^{+110}:\\
\;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\
\mathbf{if}\;b \le 1.3192368517509897 \cdot 10^{-110}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\
\mathbf{if}\;b \le 6.578629845313264 \cdot 10^{+85}:\\
\;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(c \cdot 4\right)}{(c \cdot \left(\frac{a}{b} + \frac{a}{b}\right) + \left(\left(-b\right) - b\right))_*}\\
\end{array}\]
Target
| Original | 33.7 |
|---|
| Target | 20.6 |
|---|
| Herbie | 8.6 |
|---|
\[\begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\
\end{array}\]
Derivation
- Split input into 4 regimes
if b < -1.3907652222998155e+110
Initial program 46.4
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Taylor expanded around -inf 8.4
\[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
Applied simplify2.5
\[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
if -1.3907652222998155e+110 < b < 1.3192368517509897e-110
Initial program 11.7
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
- Using strategy
rm Applied clear-num11.9
\[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]
if 1.3192368517509897e-110 < b < 6.578629845313264e+85
Initial program 41.1
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
- Using strategy
rm Applied flip-+41.2
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
Applied simplify14.9
\[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
if 6.578629845313264e+85 < b
Initial program 58.4
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
- Using strategy
rm Applied flip-+58.4
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
Applied simplify30.8
\[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
Taylor expanded around inf 14.2
\[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}}{2 \cdot a}\]
Applied simplify2.4
\[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(c \cdot 4\right)}{(c \cdot \left(\frac{a}{b} + \frac{a}{b}\right) + \left(\left(-b\right) - b\right))_*}}\]
- Recombined 4 regimes into one program.
Runtime
herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (a b c)
:name "The quadratic formula (r1)"
:herbie-target
(if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))
