2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 20.3s

Precision: 64

Math

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

↓

\[2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right)}\right)\right)\right)\]

C

double f(double x, double eps) {
        double r6370023 = x;
        double r6370024 = eps;
        double r6370025 = r6370023 + r6370024;
        double r6370026 = sin(r6370025);
        double r6370027 = sin(r6370023);
        double r6370028 = r6370026 - r6370027;
        return r6370028;
}

↓

double f(double x, double eps) {
        double r6370029 = 2.0;
        double r6370030 = eps;
        double r6370031 = 0.5;
        double r6370032 = r6370030 * r6370031;
        double r6370033 = sin(r6370032);
        double r6370034 = x;
        double r6370035 = cos(r6370034);
        double r6370036 = cos(r6370032);
        double r6370037 = r6370035 * r6370036;
        double r6370038 = r6370033 * r6370033;
        double r6370039 = r6370033 * r6370038;
        double r6370040 = sin(r6370034);
        double r6370041 = r6370040 * r6370040;
        double r6370042 = r6370040 * r6370041;
        double r6370043 = r6370039 * r6370042;
        double r6370044 = cbrt(r6370043);
        double r6370045 = r6370037 - r6370044;
        double r6370046 = log1p(r6370045);
        double r6370047 = expm1(r6370046);
        double r6370048 = r6370033 * r6370047;
        double r6370049 = r6370029 * r6370048;
        return r6370049;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.0
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 36.8

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

3. Applied diff-sin 37.2

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

4. Simplified 15.0

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

6. Applied expm1-log1p-u 15.1

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\right)\]

7. Simplified 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)}\right)\right)\]
8. Using strategy rm

9. Applied fma-udef 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\right)\right)\]

10. Applied cos-sum 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right)\right)\]
11. Using strategy rm

12. Applied add-cbrt-cube 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}}\right)\right)\right)\]

13. Applied add-cbrt-cube 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}} \cdot \sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}\right)\right)\right)\]

14. Applied cbrt-unprod 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right)}}\right)\right)\right)\]

15. Final simplification 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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)))