2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r2203061 = x;
        double r2203062 = eps;
        double r2203063 = r2203061 + r2203062;
        double r2203064 = sin(r2203063);
        double r2203065 = sin(r2203061);
        double r2203066 = r2203064 - r2203065;
        return r2203066;
}

↓

double f(double x, double eps) {
        double r2203067 = eps;
        double r2203068 = -8.778095247816633e-09;
        bool r2203069 = r2203067 <= r2203068;
        double r2203070 = x;
        double r2203071 = sin(r2203070);
        double r2203072 = cos(r2203067);
        double r2203073 = r2203071 * r2203072;
        double r2203074 = cos(r2203070);
        double r2203075 = sin(r2203067);
        double r2203076 = r2203074 * r2203075;
        double r2203077 = r2203073 + r2203076;
        double r2203078 = r2203077 - r2203071;
        double r2203079 = 1.9528216764299977e-08;
        bool r2203080 = r2203067 <= r2203079;
        double r2203081 = 2.0;
        double r2203082 = r2203067 / r2203081;
        double r2203083 = sin(r2203082);
        double r2203084 = fma(r2203081, r2203070, r2203067);
        double r2203085 = r2203084 / r2203081;
        double r2203086 = cos(r2203085);
        double r2203087 = r2203083 * r2203086;
        double r2203088 = r2203087 * r2203081;
        double r2203089 = r2203080 ? r2203088 : r2203078;
        double r2203090 = r2203069 ? r2203078 : r2203089;
        return r2203090;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.778095247816633e-09 or 1.9528216764299977e-08 < eps

   1. Initial program 30.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -8.778095247816633e-09 < eps < 1.9528216764299977e-08

   1. Initial program 44.6

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

   3. Applied diff-sin 44.6

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

   4. Simplified 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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