Average Error: 33.3 → 8.9
Time: 1.7m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l}
\mathbf{if}\;b \le -9.854202694263567 \cdot 10^{+42}:\\
\;\;\;\;\frac{\frac{4 \cdot c}{\left(\left(-b\right) - b\right) + \frac{c + c}{\frac{b}{a}}}}{2}\\
\mathbf{if}\;b \le -8.012318443752844 \cdot 10^{-94}:\\
\;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\
\mathbf{if}\;b \le 3.274275238011497 \cdot 10^{+87}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\
\end{array}\]
Target
| Original | 33.3 |
|---|
| Target | 20.6 |
|---|
| Herbie | 8.9 |
|---|
\[\begin{array}{l}
\mathbf{if}\;b \lt 0:\\
\;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\
\end{array}\]
Derivation
- Split input into 4 regimes
if b < -9.854202694263567e+42
Initial program 56.1
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied flip--56.1
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify29.1
\[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
Taylor expanded around -inf 15.7
\[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a}\]
Applied simplify4.2
\[\leadsto \color{blue}{\frac{\frac{4 \cdot c}{\left(\left(-b\right) - b\right) + \frac{c + c}{\frac{b}{a}}}}{2}}\]
if -9.854202694263567e+42 < b < -8.012318443752844e-94
Initial program 40.8
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied flip--40.9
\[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
Applied simplify15.4
\[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
if -8.012318443752844e-94 < b < 3.274275238011497e+87
Initial program 12.3
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
- Using strategy
rm Applied div-sub12.3
\[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\]
if 3.274275238011497e+87 < b
Initial program 41.9
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Taylor expanded around inf 8.9
\[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
Applied simplify3.8
\[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
- Recombined 4 regimes into one program.
Runtime
herbie shell --seed '#(1151762963 887253659 3096734101 777879090 2714024476 786371635)'
(FPCore (a b c)
:name "The quadratic formula (r2)"
: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)))
