Average Error: 33.3 → 9.1
Time: 1.7m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l}
\mathbf{if}\;b \le -5326805461.975518:\\
\;\;\;\;\frac{\frac{4 \cdot c}{\left(\left(-b\right) - b\right) + \frac{a}{b} \cdot \left(c + c\right)}}{2}\\
\mathbf{if}\;b \le -1.431414383204406 \cdot 10^{-78}:\\
\;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\
\mathbf{if}\;b \le -5.863009078075828 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{4 \cdot c}{\left(\left(-b\right) - b\right) + \frac{a}{b} \cdot \left(c + c\right)}}{2}\\
\mathbf{if}\;b \le 2.150797236902381 \cdot 10^{+151}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{-b}{a}\\
\end{array}\]
Target
| Original | 33.3 |
|---|
| Target | 20.1 |
|---|
| Herbie | 9.1 |
|---|
\[\begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\end{array}\]
Derivation
- Split input into 4 regimes
if b < -5326805461.975518 or -1.431414383204406e-78 < b < -5.863009078075828e-98
Initial program 54.7
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied flip--54.7
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify26.7
\[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
Taylor expanded around -inf 16.6
\[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a}\]
Applied simplify6.1
\[\leadsto \color{blue}{\frac{\frac{4 \cdot c}{\left(\left(-b\right) - b\right) + \frac{a}{b} \cdot \left(c + c\right)}}{2}}\]
if -5326805461.975518 < b < -1.431414383204406e-78
Initial program 40.2
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied flip--40.3
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify16.8
\[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
if -5.863009078075828e-98 < b < 2.150797236902381e+151
Initial program 11.5
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied div-sub11.5
\[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\]
if 2.150797236902381e+151 < b
Initial program 59.0
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Taylor expanded around inf 2.4
\[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
Applied simplify2.4
\[\leadsto \color{blue}{\frac{-b}{a}}\]
- Recombined 4 regimes into one program.
Runtime
herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)'
(FPCore (a b c)
:name "quadm (p42, negative)"
:herbie-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)))
