Average Error: 36.8 → 0.4
Time: 6.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r121575 = x;
        double r121576 = eps;
        double r121577 = r121575 + r121576;
        double r121578 = sin(r121577);
        double r121579 = sin(r121575);
        double r121580 = r121578 - r121579;
        return r121580;
}
↓
double f(double x, double eps) {
        double r121581 = x;
        double r121582 = sin(r121581);
        double r121583 = eps;
        double r121584 = cos(r121583);
        double r121585 = 3.0;
        double r121586 = pow(r121584, r121585);
        double r121587 = 1.0;
        double r121588 = r121586 - r121587;
        double r121589 = r121584 + r121587;
        double r121590 = r121584 * r121589;
        double r121591 = r121590 + r121587;
        double r121592 = r121588 / r121591;
        double r121593 = r121582 * r121592;
        double r121594 = cos(r121581);
        double r121595 = sin(r121583);
        double r121596 = r121594 * r121595;
        double r121597 = r121593 + r121596;
        return r121597;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied sin-sum21.7

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

  4. Taylor expanded around inf 21.7

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

  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 flip3--0.4

    \[\leadsto \sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\]

  8. Simplified0.4

    \[\leadsto \sin x \cdot \frac{\color{blue}{{\left(\cos \varepsilon\right)}^{3} - 1}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]

  9. Simplified0.4

    \[\leadsto \sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1}} + \cos x \cdot \sin \varepsilon\]

  10. Final simplification0.4

    \[\leadsto \sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020089 
(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)))