2sin (example 3.3)

Average Error: 36.9 → 0.4

Time: 5.9s

Precision: 64

Math

\[\sin \left(x + \varepsilon\right) - \sin x\]

↓

\[\mathsf{fma}\left(\sin x, \frac{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos \varepsilon\right)}^{3} - 1\right)\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]

C

double f(double x, double eps) {
        double r108084 = x;
        double r108085 = eps;
        double r108086 = r108084 + r108085;
        double r108087 = sin(r108086);
        double r108088 = sin(r108084);
        double r108089 = r108087 - r108088;
        return r108089;
}

↓

double f(double x, double eps) {
        double r108090 = x;
        double r108091 = sin(r108090);
        double r108092 = eps;
        double r108093 = cos(r108092);
        double r108094 = 3.0;
        double r108095 = pow(r108093, r108094);
        double r108096 = 1.0;
        double r108097 = r108095 - r108096;
        double r108098 = expm1(r108097);
        double r108099 = log1p(r108098);
        double r108100 = r108093 + r108096;
        double r108101 = fma(r108093, r108100, r108096);
        double r108102 = r108099 / r108101;
        double r108103 = cos(r108090);
        double r108104 = sin(r108092);
        double r108105 = r108103 * r108104;
        double r108106 = fma(r108091, r108102, r108105);
        return r108106;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.3
Herbie	0.4

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

Derivation

1. Initial program 36.9

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

3. Applied sin-sum 21.4

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

4. Taylor expanded around inf 21.4

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

5. Simplified 0.4

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

7. Applied flip3-- 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

   \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}, \cos x \cdot \sin \varepsilon\right)\]
10. Using strategy rm

11. Applied log1p-expm1-u 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos \varepsilon\right)}^{3} - 1\right)\right)}}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]

12. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos \varepsilon\right)}^{3} - 1\right)\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(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)))