Average Error: 37.2 → 0.4
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r111543 = x;
        double r111544 = eps;
        double r111545 = r111543 + r111544;
        double r111546 = sin(r111545);
        double r111547 = sin(r111543);
        double r111548 = r111546 - r111547;
        return r111548;
}
↓
double f(double x, double eps) {
        double r111549 = x;
        double r111550 = sin(r111549);
        double r111551 = eps;
        double r111552 = sin(r111551);
        double r111553 = r111552 * r111552;
        double r111554 = -r111553;
        double r111555 = r111550 * r111554;
        double r111556 = cos(r111551);
        double r111557 = 1.0;
        double r111558 = r111556 + r111557;
        double r111559 = r111555 / r111558;
        double r111560 = cos(r111549);
        double r111561 = r111560 * r111552;
        double r111562 = r111559 + r111561;
        return r111562;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied sin-sum22.1

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

  5. Applied *-un-lft-identity22.1

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

  6. Applied *-un-lft-identity22.1

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

  7. Applied distribute-lft-out--22.1

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

  8. Simplified0.4

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

  10. Applied flip--0.5

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

  11. Applied associate-*r/0.5

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019352 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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