Average Error: 36.8 → 0.3
Time: 21.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \left(\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) + \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right) \cdot \left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \left(\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) + \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right) \cdot \left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right)\right)
double f(double x, double eps) {
        double r5608388 = x;
        double r5608389 = eps;
        double r5608390 = r5608388 + r5608389;
        double r5608391 = sin(r5608390);
        double r5608392 = sin(r5608388);
        double r5608393 = r5608391 - r5608392;
        return r5608393;
}
↓
double f(double x, double eps) {
        double r5608394 = eps;
        double r5608395 = 2.0;
        double r5608396 = r5608394 / r5608395;
        double r5608397 = sin(r5608396);
        double r5608398 = -2.0;
        double r5608399 = r5608397 * r5608398;
        double r5608400 = x;
        double r5608401 = sin(r5608400);
        double r5608402 = r5608401 * r5608397;
        double r5608403 = r5608399 * r5608402;
        double r5608404 = r5608397 * r5608395;
        double r5608405 = cos(r5608400);
        double r5608406 = cos(r5608396);
        double r5608407 = r5608405 * r5608406;
        double r5608408 = r5608404 * r5608407;
        double r5608409 = r5608403 + r5608408;
        return r5608409;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.9
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied diff-sin37.1

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied cos-sum0.3

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

  10. Applied sub-neg0.3

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

  11. Applied distribute-lft-in0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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