Average Error: 37.1 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r118383 = x;
        double r118384 = eps;
        double r118385 = r118383 + r118384;
        double r118386 = sin(r118385);
        double r118387 = sin(r118383);
        double r118388 = r118386 - r118387;
        return r118388;
}
↓
double f(double x, double eps) {
        double r118389 = x;
        double r118390 = sin(r118389);
        double r118391 = eps;
        double r118392 = cos(r118391);
        double r118393 = 1.0;
        double r118394 = r118392 - r118393;
        double r118395 = r118390 * r118394;
        double r118396 = exp(r118395);
        double r118397 = log(r118396);
        double r118398 = cos(r118389);
        double r118399 = sin(r118391);
        double r118400 = r118398 * r118399;
        double r118401 = r118397 + r118400;
        return r118401;
}

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

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum21.5

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

  4. Applied associate--l+21.5

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

  5. Taylor expanded around inf 21.5

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

  6. Simplified0.4

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

  8. Applied add-log-exp0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019344 
(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)))