2sin (example 3.3)

Average Error: 37.4 → 0.5

Time: 24.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0987473663124097:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.300741008615541 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r2984396 = x;
        double r2984397 = eps;
        double r2984398 = r2984396 + r2984397;
        double r2984399 = sin(r2984398);
        double r2984400 = sin(r2984396);
        double r2984401 = r2984399 - r2984400;
        return r2984401;
}

↓

double f(double x, double eps) {
        double r2984402 = eps;
        double r2984403 = -0.0987473663124097;
        bool r2984404 = r2984402 <= r2984403;
        double r2984405 = x;
        double r2984406 = sin(r2984405);
        double r2984407 = cos(r2984402);
        double r2984408 = r2984406 * r2984407;
        double r2984409 = cos(r2984405);
        double r2984410 = sin(r2984402);
        double r2984411 = r2984409 * r2984410;
        double r2984412 = r2984408 + r2984411;
        double r2984413 = r2984412 - r2984406;
        double r2984414 = 1.300741008615541e-08;
        bool r2984415 = r2984402 <= r2984414;
        double r2984416 = 2.0;
        double r2984417 = r2984402 / r2984416;
        double r2984418 = sin(r2984417);
        double r2984419 = r2984405 + r2984402;
        double r2984420 = r2984405 + r2984419;
        double r2984421 = r2984420 / r2984416;
        double r2984422 = cos(r2984421);
        double r2984423 = r2984418 * r2984422;
        double r2984424 = r2984416 * r2984423;
        double r2984425 = r2984415 ? r2984424 : r2984413;
        double r2984426 = r2984404 ? r2984413 : r2984425;
        return r2984426;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	14.9
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.0987473663124097 or 1.300741008615541e-08 < eps

   1. Initial program 30.0

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

   3. Applied sin-sum 0.5

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

   if -0.0987473663124097 < eps < 1.300741008615541e-08

   1. Initial program 44.8

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

   3. Applied diff-sin 44.8

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

   4. Simplified 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0987473663124097:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.300741008615541 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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