Average Error: 37.1 → 0.4
Time: 15.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot \sin x\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot \sin x
double f(double x, double eps) {
        double r103065 = x;
        double r103066 = eps;
        double r103067 = r103065 + r103066;
        double r103068 = sin(r103067);
        double r103069 = sin(r103065);
        double r103070 = r103068 - r103069;
        return r103070;
}
↓
double f(double x, double eps) {
        double r103071 = x;
        double r103072 = cos(r103071);
        double r103073 = eps;
        double r103074 = sin(r103073);
        double r103075 = r103072 * r103074;
        double r103076 = cos(r103073);
        double r103077 = 1.0;
        double r103078 = r103076 - r103077;
        double r103079 = sin(r103071);
        double r103080 = r103078 * r103079;
        double r103081 = r103075 + r103080;
        return r103081;
}

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
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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 add-log-exp0.4

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

  11. Applied add-log-exp0.4

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

  12. Applied diff-log0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(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)))