Average Error: 35.6 → 28.7
Time: 38.6s
Precision: 64
Internal Precision: 2368
$y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)$
$\begin{array}{l} \mathbf{if}\;y \cdot \sin \left(\left(\sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}} \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \le -8.61835313756233 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \sin \left(\frac{\frac{\pi}{4}}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \cdot \sin \left(\left(\sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}} \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \le 3.7419603906382095 \cdot 10^{-293}:\\ \;\;\;\;x \cdot \left(\pi \cdot \frac{1}{4}\right)\\ \mathbf{if}\;y \cdot \sin \left(\left(\sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}} \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \cdot \sqrt[3]{\frac{\frac{\pi}{4} \cdot x}{y}}\right) \le 1.2902542149373409 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)} \cdot \sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)}\right) \cdot \sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 4 regimes
2. ## if (* y (sin (* (* (cbrt (/ (* (/ PI 4) x) y)) (cbrt (/ (* (/ PI 4) x) y))) (cbrt (/ (* (/ PI 4) x) y))))) < -8.61835313756233e-309

1. Initial program 27.2

$y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)$
2. Using strategy rm
3. Applied associate-/l*28.0

$\leadsto y \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{4}}{\frac{y}{x}}\right)}$

## if -8.61835313756233e-309 < (* y (sin (* (* (cbrt (/ (* (/ PI 4) x) y)) (cbrt (/ (* (/ PI 4) x) y))) (cbrt (/ (* (/ PI 4) x) y))))) < 3.7419603906382095e-293

1. Initial program 55.6

$y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)$
2. Taylor expanded around inf 56.1

$\leadsto y \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{x \cdot \pi}{y}\right)}$
3. Applied simplify3.2

$\leadsto \color{blue}{x \cdot \left(\pi \cdot \frac{1}{4}\right)}$

## if 3.7419603906382095e-293 < (* y (sin (* (* (cbrt (/ (* (/ PI 4) x) y)) (cbrt (/ (* (/ PI 4) x) y))) (cbrt (/ (* (/ PI 4) x) y))))) < 1.2902542149373409e+240

1. Initial program 27.6

$y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)$
2. Using strategy rm

$\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)} \cdot \sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)}\right) \cdot \sqrt[3]{y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)}}$

## if 1.2902542149373409e+240 < (* y (sin (* (* (cbrt (/ (* (/ PI 4) x) y)) (cbrt (/ (* (/ PI 4) x) y))) (cbrt (/ (* (/ PI 4) x) y)))))

1. Initial program 61.9

$y \cdot \sin \left(\frac{\frac{\pi}{4} \cdot x}{y}\right)$
2. Taylor expanded around 0 59.9

$\leadsto y \cdot \color{blue}{0}$
3. Applied simplify59.9

$\leadsto \color{blue}{0}$
3. Recombined 4 regimes into one program.

# Runtime

Time bar (total: 38.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (y x)
:name "y*sin(PI/4*x/y)"
(* y (sin (/ (* (/ PI 4) x) y))))