Average Error: 37.0 → 0.3
Time: 24.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot 2\right)
double f(double x, double eps) {
        double r12408677 = x;
        double r12408678 = eps;
        double r12408679 = r12408677 + r12408678;
        double r12408680 = sin(r12408679);
        double r12408681 = sin(r12408677);
        double r12408682 = r12408680 - r12408681;
        return r12408682;
}
↓
double f(double x, double eps) {
        double r12408683 = 0.5;
        double r12408684 = eps;
        double r12408685 = r12408683 * r12408684;
        double r12408686 = cos(r12408685);
        double r12408687 = x;
        double r12408688 = cos(r12408687);
        double r12408689 = r12408686 * r12408688;
        double r12408690 = sin(r12408687);
        double r12408691 = sin(r12408685);
        double r12408692 = r12408690 * r12408691;
        double r12408693 = r12408689 - r12408692;
        double r12408694 = 2.0;
        double r12408695 = r12408691 * r12408694;
        double r12408696 = r12408693 * r12408695;
        return r12408696;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied diff-sin37.4

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied cos-sum0.3

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

  10. Applied *-commutative0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019173 
(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)))