Average Error: 37.1 → 0.3
Time: 5.8s
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 r91000 = x;
        double r91001 = eps;
        double r91002 = r91000 + r91001;
        double r91003 = sin(r91002);
        double r91004 = sin(r91000);
        double r91005 = r91003 - r91004;
        return r91005;
}
↓
double f(double x, double eps) {
        double r91006 = x;
        double r91007 = sin(r91006);
        double r91008 = eps;
        double r91009 = sin(r91008);
        double r91010 = r91009 * r91009;
        double r91011 = -r91010;
        double r91012 = r91007 * r91011;
        double r91013 = cos(r91008);
        double r91014 = 1.0;
        double r91015 = r91013 + r91014;
        double r91016 = r91012 / r91015;
        double r91017 = cos(r91006);
        double r91018 = r91017 * r91009;
        double r91019 = r91016 + r91018;
        return r91019;
}

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
Target14.4
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum22.6

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

  4. Taylor expanded around inf 22.6

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

  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 flip--0.5

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

  8. Applied associate-*r/0.5

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

  9. Simplified0.5

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

  11. Applied sub-1-cos0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020035 
(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)))