Average Error: 36.0 → 29.0

Time: 7.3s

Precision: binary64

\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

↓

\[1\]

\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}

↓

(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))

↓

(FPCore (x y) :precision binary64 1.0)

double code(double x, double y) {
	return (((double) tan((x / ((double) (y * 2.0))))) / ((double) sin((x / ((double) (y * 2.0))))));
}

↓

double code(double x, double y) {
	return 1.0;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	36.0
Target	29.5
Herbie	29.0

\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

Initial program Error: 36.0 bits
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
Taylor expanded around 0 Error: 29.0 bits
\[\leadsto \color{blue}{1}\]
Final simplificationError: 29.0 bits
\[\leadsto 1\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))