Average Error: 36.8 → 0.3
Time: 36.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right) + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\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(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)
double f(double x, double eps) {
        double r5324351 = x;
        double r5324352 = eps;
        double r5324353 = r5324351 + r5324352;
        double r5324354 = sin(r5324353);
        double r5324355 = sin(r5324351);
        double r5324356 = r5324354 - r5324355;
        return r5324356;
}
↓
double f(double x, double eps) {
        double r5324357 = 0.5;
        double r5324358 = eps;
        double r5324359 = r5324357 * r5324358;
        double r5324360 = cos(r5324359);
        double r5324361 = x;
        double r5324362 = cos(r5324361);
        double r5324363 = r5324360 * r5324362;
        double r5324364 = sin(r5324359);
        double r5324365 = 2.0;
        double r5324366 = r5324364 * r5324365;
        double r5324367 = r5324363 * r5324366;
        double r5324368 = -2.0;
        double r5324369 = r5324368 * r5324364;
        double r5324370 = sin(r5324361);
        double r5324371 = r5324364 * r5324370;
        double r5324372 = r5324369 * r5324371;
        double r5324373 = r5324367 + r5324372;
        return r5324373;
}

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

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied cos-sum0.3

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

  11. Applied sub-neg0.3

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

  12. Applied distribute-lft-in0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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)))