Average Error: 32.4 → 16.2
Time: 34.7s
Precision: 64
Internal Precision: 576
$\sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}$
$\begin{array}{l} \mathbf{if}\;\cos t \cdot \left(-x\right) \le -2.334906730441395 \cdot 10^{-137}:\\ \;\;\;\;\cos t \cdot x\\ \mathbf{if}\;\cos t \cdot \left(-x\right) \le 6.073623898151345 \cdot 10^{-108}:\\ \;\;\;\;\left|\sin t \cdot y\right| + \frac{1}{2} \cdot \frac{{x}^{2}}{\left|\sin t \cdot y\right|}\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot \left(-x\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if (* (cos t) (- x)) < -2.334906730441395e-137

1. Initial program 31.3

$\sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}$
2. Taylor expanded around inf 16.3

$\leadsto \color{blue}{\cos t \cdot x}$

## if -2.334906730441395e-137 < (* (cos t) (- x)) < 6.073623898151345e-108

1. Initial program 34.2

$\sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}$
2. Using strategy rm

$\leadsto \sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \color{blue}{\sqrt{\left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t} \cdot \sqrt{\left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}}}$
4. Applied simplify34.2

$\leadsto \sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \color{blue}{\left|\sin t \cdot y\right|} \cdot \sqrt{\left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}}$
5. Applied simplify28.1

$\leadsto \sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \left|\sin t \cdot y\right| \cdot \color{blue}{\left|\sin t \cdot y\right|}}$
6. Taylor expanded around 0 17.1

$\leadsto \color{blue}{\left|\sin t \cdot y\right| + \frac{1}{2} \cdot \frac{{x}^{2}}{\left|\sin t \cdot y\right|}}$

## if 6.073623898151345e-108 < (* (cos t) (- x))

1. Initial program 31.9

$\sqrt{\left(\left(x \cdot x\right) \cdot \cos t\right) \cdot \cos t + \left(\left(y \cdot y\right) \cdot \sin t\right) \cdot \sin t}$
2. Taylor expanded around -inf 15.2

$\leadsto \color{blue}{-1 \cdot \left(\cos t \cdot x\right)}$
3. Applied simplify15.2

$\leadsto \color{blue}{\cos t \cdot \left(-x\right)}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 34.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x t y)
:name "sqrt(x*x*cos(t)*cos(t)+y*y*sin(t)*sin(t))"
(sqrt (+ (* (* (* x x) (cos t)) (cos t)) (* (* (* y y) (sin t)) (sin t)))))