Average Error: 29.8 → 3.6
Time: 6.1s
Precision: 64
Internal Precision: 2368
$b + \sqrt{\left(-b\right) \cdot \left(-b\right) - c}$
$\begin{array}{l} \mathbf{if}\;b \le -2.2869001147814792 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot c}{b}\\ \mathbf{elif}\;b \le 4.412319935246271 \cdot 10^{+79}:\\ \;\;\;\;b + \sqrt{b \cdot b - c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 2 - \frac{c}{b} \cdot \frac{1}{2}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if b < -2.2869001147814792e-45

1. Initial program 52.7

$b + \sqrt{\left(-b\right) \cdot \left(-b\right) - c}$
2. Initial simplification52.7

$\leadsto b + \sqrt{b \cdot b - c}$
3. Taylor expanded around -inf 5.6

$\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b}}$
4. Using strategy rm
5. Applied associate-*r/5.6

$\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}}$

## if -2.2869001147814792e-45 < b < 4.412319935246271e+79

1. Initial program 3.0

$b + \sqrt{\left(-b\right) \cdot \left(-b\right) - c}$
2. Initial simplification3.0

$\leadsto b + \sqrt{b \cdot b - c}$

## if 4.412319935246271e+79 < b

1. Initial program 41.0

$b + \sqrt{\left(-b\right) \cdot \left(-b\right) - c}$
2. Initial simplification41.0

$\leadsto b + \sqrt{b \cdot b - c}$
3. Taylor expanded around inf 1.6

$\leadsto \color{blue}{2 \cdot b - \frac{1}{2} \cdot \frac{c}{b}}$
3. Recombined 3 regimes into one program.
4. Final simplification3.6

$\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2869001147814792 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot c}{b}\\ \mathbf{elif}\;b \le 4.412319935246271 \cdot 10^{+79}:\\ \;\;\;\;b + \sqrt{b \cdot b - c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 2 - \frac{c}{b} \cdot \frac{1}{2}\\ \end{array}$

# Runtime

Time bar (total: 6.1s)Debug log

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