2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 1.5m

Precision: 64

Math

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

↓

\[\left(\left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]

C

double f(double x, double eps) {
        double r6993186 = x;
        double r6993187 = eps;
        double r6993188 = r6993186 + r6993187;
        double r6993189 = sin(r6993188);
        double r6993190 = sin(r6993186);
        double r6993191 = r6993189 - r6993190;
        return r6993191;
}

↓

double f(double x, double eps) {
        double r6993192 = x;
        double r6993193 = cos(r6993192);
        double r6993194 = eps;
        double r6993195 = 2.0;
        double r6993196 = r6993194 / r6993195;
        double r6993197 = cos(r6993196);
        double r6993198 = r6993193 * r6993197;
        double r6993199 = sin(r6993196);
        double r6993200 = sin(r6993192);
        double r6993201 = r6993199 * r6993200;
        double r6993202 = r6993198 - r6993201;
        double r6993203 = 0.5;
        double r6993204 = r6993203 * r6993194;
        double r6993205 = sin(r6993204);
        double r6993206 = r6993202 * r6993205;
        double r6993207 = r6993206 * r6993195;
        return r6993207;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.0
Herbie	0.3

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

Derivation

1. Initial program 37.1

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

3. Applied diff-sin 37.4

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

4. Simplified 15.0

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)}\]
5. Using strategy rm

6. Applied add-log-exp 15.2

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}\right)}\right)\]

7. Simplified 15.2

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \log \color{blue}{\left(e^{\cos \left(x + \frac{\varepsilon}{2}\right)}\right)}\right)\]
8. Using strategy rm

9. Applied cos-sum 0.6

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \log \left(e^{\color{blue}{\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}}\right)\right)\]

10. Applied exp-diff 0.6

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \log \color{blue}{\left(\frac{e^{\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right)}}{e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}}\right)}\right)\]

11. Applied log-div 0.6

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\log \left(e^{\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right)}\right) - \log \left(e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right)}\right)\]

12. Simplified 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{\cos \left(\frac{\varepsilon}{2}\right) \cdot \cos x} - \log \left(e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right)\right)\]

13. Simplified 0.3

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \cos x - \color{blue}{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right)\]

14. Final simplification 0.3

    \[\leadsto \left(\left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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