2sin (example 3.3)

Average Error: 36.7 → 0.3

Time: 16.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r76146 = x;
        double r76147 = eps;
        double r76148 = r76146 + r76147;
        double r76149 = sin(r76148);
        double r76150 = sin(r76146);
        double r76151 = r76149 - r76150;
        return r76151;
}

↓

double f(double x, double eps) {
        double r76152 = 2.0;
        double r76153 = 0.5;
        double r76154 = eps;
        double r76155 = r76153 * r76154;
        double r76156 = sin(r76155);
        double r76157 = cos(r76155);
        double r76158 = x;
        double r76159 = cos(r76158);
        double r76160 = -r76156;
        double r76161 = sin(r76158);
        double r76162 = r76160 * r76161;
        double r76163 = fma(r76157, r76159, r76162);
        double r76164 = r76156 * r76163;
        double r76165 = r76152 * r76164;
        return r76165;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	15.0
Herbie	0.3

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

Derivation

1. Initial program 36.7

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

3. Applied diff-sin 37.0

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

4. Simplified 15.1

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

5. Taylor expanded around inf 15.0

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

6. Simplified 15.0

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

8. Applied fma-udef 15.0

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

9. Applied cos-sum 0.3

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

11. Applied fma-neg 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019198 +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)))