Average Error: 25.2 → 7.7
Time: 15.5s
Precision: 64
Internal Precision: 576
$\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}$
$\begin{array}{l} \mathbf{if}\;b \le -6.826205091476792 \cdot 10^{+149}:\\ \;\;\;\;\frac{a \cdot c}{b} \cdot 2 - b\\ \mathbf{elif}\;b \le 1.8905452877783028 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;b - \frac{a \cdot c}{b} \cdot 2\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if b < -6.826205091476792e+149

1. Initial program 57.9

$\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}$
2. Initial simplification57.9

$\leadsto \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}$
3. Taylor expanded around -inf 6.4

$\leadsto \color{blue}{2 \cdot \frac{a \cdot c}{b} - b}$

## if -6.826205091476792e+149 < b < 1.8905452877783028e+140

1. Initial program 8.4

$\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}$
2. Initial simplification8.4

$\leadsto \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}$

## if 1.8905452877783028e+140 < b

1. Initial program 55.8

$\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}$
2. Initial simplification55.8

$\leadsto \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}$
3. Taylor expanded around inf 6.5

$\leadsto \color{blue}{b - 2 \cdot \frac{a \cdot c}{b}}$
3. Recombined 3 regimes into one program.
4. Final simplification7.7

$\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.826205091476792 \cdot 10^{+149}:\\ \;\;\;\;\frac{a \cdot c}{b} \cdot 2 - b\\ \mathbf{elif}\;b \le 1.8905452877783028 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;b - \frac{a \cdot c}{b} \cdot 2\\ \end{array}$

# Runtime

Time bar (total: 15.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (b a c)
:name "sqrt(b * b - 4 * a * c)"
(sqrt (- (* b b) (* (* 4 a) c))))