2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 6.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r96018 = x;
        double r96019 = eps;
        double r96020 = r96018 + r96019;
        double r96021 = sin(r96020);
        double r96022 = sin(r96018);
        double r96023 = r96021 - r96022;
        return r96023;
}

↓

double f(double x, double eps) {
        double r96024 = x;
        double r96025 = sin(r96024);
        double r96026 = eps;
        double r96027 = cos(r96026);
        double r96028 = 3.0;
        double r96029 = pow(r96027, r96028);
        double r96030 = 1.0;
        double r96031 = r96029 - r96030;
        double r96032 = r96027 + r96030;
        double r96033 = fma(r96027, r96032, r96030);
        double r96034 = r96031 / r96033;
        double r96035 = expm1(r96034);
        double r96036 = log1p(r96035);
        double r96037 = cos(r96024);
        double r96038 = sin(r96026);
        double r96039 = r96037 * r96038;
        double r96040 = fma(r96025, r96036, r96039);
        return r96040;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
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 37.1

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

3. Applied sin-sum 21.9

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

5. Applied *-un-lft-identity 21.9

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

6. Applied *-un-lft-identity 21.9

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

7. Applied distribute-lft-out-- 21.9

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

8. Simplified 0.4

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

10. Applied log1p-expm1-u 0.4

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right)}, \cos x \cdot \sin \varepsilon\right)\]
11. Using strategy rm

12. Applied flip3-- 0.4

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

13. Simplified 0.4

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

14. Simplified 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020046 +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)))