Average Error: 36.9 → 0.4
Time: 9.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos \varepsilon - 1\right) \cdot \sin x + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\cos \varepsilon - 1\right) \cdot \sin x + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r486392 = x;
        double r486393 = eps;
        double r486394 = r486392 + r486393;
        double r486395 = sin(r486394);
        double r486396 = sin(r486392);
        double r486397 = r486395 - r486396;
        return r486397;
}
↓
double f(double x, double eps) {
        double r486398 = eps;
        double r486399 = cos(r486398);
        double r486400 = 1.0;
        double r486401 = r486399 - r486400;
        double r486402 = x;
        double r486403 = sin(r486402);
        double r486404 = r486401 * r486403;
        double r486405 = cos(r486402);
        double r486406 = sin(r486398);
        double r486407 = r486405 * r486406;
        double r486408 = r486404 + r486407;
        return r486408;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
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 36.9

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

  3. Applied sin-sum21.8

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

  4. Taylor expanded around inf 21.8

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

  5. Simplified0.4

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

  7. Applied add-log-exp0.4

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

  8. Applied add-log-exp0.4

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

  9. Applied diff-log0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 350497007 
(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)))