Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.1 → 0.1

Time: 4.6s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y) {
	return ((double) (((double) (((double) sin(x)) * ((double) sinh(y)))) / x));
}

↓

double code(double x, double y) {
	return ((double) (((double) sinh(y)) * ((double) (((double) sin(x)) / x))));
}

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 0.4

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

9. Applied sinh-def 43.7

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

10. Applied associate-*l/ 43.7

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

11. Applied associate-*r/ 43.7

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

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 2020173 
(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))