Average Error: 37.1 → 0.3
Time: 25.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r11959367 = x;
        double r11959368 = eps;
        double r11959369 = r11959367 + r11959368;
        double r11959370 = sin(r11959369);
        double r11959371 = sin(r11959367);
        double r11959372 = r11959370 - r11959371;
        return r11959372;
}
↓
double f(double x, double eps) {
        double r11959373 = 0.5;
        double r11959374 = eps;
        double r11959375 = r11959373 * r11959374;
        double r11959376 = sin(r11959375);
        double r11959377 = 2.0;
        double r11959378 = r11959376 * r11959377;
        double r11959379 = cos(r11959375);
        double r11959380 = x;
        double r11959381 = cos(r11959380);
        double r11959382 = r11959379 * r11959381;
        double r11959383 = r11959378 * r11959382;
        double r11959384 = -2.0;
        double r11959385 = r11959384 * r11959376;
        double r11959386 = sin(r11959380);
        double r11959387 = r11959386 * r11959376;
        double r11959388 = r11959385 * r11959387;
        double r11959389 = r11959383 + r11959388;
        return r11959389;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.1
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied diff-sin37.4

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied cos-sum0.3

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

  10. Applied sub-neg0.3

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

  11. Applied distribute-rgt-in0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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