2sin (example 3.3)

Average Error: 37.1 → 0.7

Time: 5.6s

Precision: binary64

Math

\[\sin \left(x + \varepsilon\right) - \sin x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001084741578760162 \lor \neg \left(\varepsilon \leq 1.7278717375445562 \cdot 10^{-25}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]

C

double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}

↓

double code(double x, double eps) {
	double VAR;
	if (((eps <= -0.0001084741578760162) || !(eps <= 1.7278717375445562e-25))) {
		VAR = ((double) (((double) (((double) (((double) sin(x)) * ((double) cos(eps)))) + ((double) (((double) cos(x)) * ((double) sin(eps)))))) - ((double) sin(x))));
	} else {
		VAR = ((double) (2.0 * ((double) (((double) sin((eps / 2.0))) * ((double) cos((((double) (x + ((double) (eps + x)))) / 2.0)))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.8
Herbie	0.7

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

1. Split input into 2 regimes

2. if eps < -1.0847415787601619e-4 or 1.72787173754455624e-25 < eps

   1. Initial program 29.5

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   2. Using strategy rm

   3. Applied sin-sum 1.2

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

   if -1.0847415787601619e-4 < eps < 1.72787173754455624e-25

   1. Initial program 45.5

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   2. Using strategy rm

   3. Applied diff-sin 45.5

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

   4. Simplified 0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001084741578760162 \lor \neg \left(\varepsilon \leq 1.7278717375445562 \cdot 10^{-25}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))