Average Error: 37.5 → 0.4
Time: 17.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \left(\cos x - \frac{\sin x}{\cos \varepsilon + 1} \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \left(\cos x - \frac{\sin x}{\cos \varepsilon + 1} \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r116364 = x;
        double r116365 = eps;
        double r116366 = r116364 + r116365;
        double r116367 = sin(r116366);
        double r116368 = sin(r116364);
        double r116369 = r116367 - r116368;
        return r116369;
}
↓
double f(double x, double eps) {
        double r116370 = eps;
        double r116371 = sin(r116370);
        double r116372 = x;
        double r116373 = cos(r116372);
        double r116374 = sin(r116372);
        double r116375 = cos(r116370);
        double r116376 = 1.0;
        double r116377 = r116375 + r116376;
        double r116378 = r116374 / r116377;
        double r116379 = r116378 * r116371;
        double r116380 = r116373 - r116379;
        double r116381 = r116371 * r116380;
        return r116381;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.5

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Simplified0.5

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019294 
(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)))