Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.1 → 0.1

Time: 4.9s

Precision: 64

Math

\[\frac{\sin x \cdot \sinh y}{x}\]

↓

\[\sinh y \cdot \frac{\sin x}{x}\]

C

double f(double x, double y) {
        double r521370 = x;
        double r521371 = sin(r521370);
        double r521372 = y;
        double r521373 = sinh(r521372);
        double r521374 = r521371 * r521373;
        double r521375 = r521374 / r521370;
        return r521375;
}

↓

double f(double x, double y) {
        double r521376 = y;
        double r521377 = sinh(r521376);
        double r521378 = x;
        double r521379 = sin(r521378);
        double r521380 = r521379 / r521378;
        double r521381 = r521377 * r521380;
        return r521381;
}

Error

Bits error versus x

Bits error versus y

Target

Original	14.1
Target	0.2
Herbie	0.1

\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

1. Initial program 14.1

   \[\frac{\sin x \cdot \sinh y}{x}\]
2. Using strategy rm

3. Applied *-un-lft-identity 14.1

   \[\leadsto \frac{\sin x \cdot \sinh y}{\color{blue}{1 \cdot x}}\]

4. Applied times-frac 0.2

   \[\leadsto \color{blue}{\frac{\sin x}{1} \cdot \frac{\sinh y}{x}}\]

5. Simplified 0.2

   \[\leadsto \color{blue}{\sin x} \cdot \frac{\sinh y}{x}\]
6. Using strategy rm

7. Applied div-inv 0.3

   \[\leadsto \sin x \cdot \color{blue}{\left(\sinh y \cdot \frac{1}{x}\right)}\]
8. Using strategy rm

9. Applied sinh-def 43.2

   \[\leadsto \sin x \cdot \left(\color{blue}{\frac{e^{y} - e^{-y}}{2}} \cdot \frac{1}{x}\right)\]

10. Applied associate-*l/ 43.2

    \[\leadsto \sin x \cdot \color{blue}{\frac{\left(e^{y} - e^{-y}\right) \cdot \frac{1}{x}}{2}}\]

11. Applied associate-*r/ 43.2

    \[\leadsto \color{blue}{\frac{\sin x \cdot \left(\left(e^{y} - e^{-y}\right) \cdot \frac{1}{x}\right)}{2}}\]

12. Simplified 0.2

    \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \frac{1}{x}\right) \cdot \left(2 \cdot \sinh y\right)}}{2}\]
13. Using strategy rm

14. Applied *-un-lft-identity 0.2

    \[\leadsto \frac{\left(\sin x \cdot \frac{1}{x}\right) \cdot \left(2 \cdot \sinh y\right)}{\color{blue}{1 \cdot 2}}\]

15. Applied times-frac 0.2

    \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{1}{x}}{1} \cdot \frac{2 \cdot \sinh y}{2}}\]

16. Simplified 0.1

    \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \frac{2 \cdot \sinh y}{2}\]

17. Simplified 0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\sinh y}{1}}\]

18. Final simplification 0.1

    \[\leadsto \sinh y \cdot \frac{\sin x}{x}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))