Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 13.8 → 0.2

Time: 4.2s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double x, double y) {
	return (((sin(x) * (1.0 / x)) * (2.0 * sinh(y))) / 2.0);
}

Error

Bits error versus x

Bits error versus y

Target

Original	13.8
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 13.8

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

3. Applied *-un-lft-identity 13.8

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

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 43.3

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

10. Applied associate-*l/ 43.3

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

11. Applied associate-*r/ 43.3

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

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020092 +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))