Linear.Quaternion:$csinh from linear-1.19.1.3

Average Error: 0.1 → 0.2

Time: 4.1s

Precision: 64

Math

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

↓

\[\frac{-\sin y}{-\frac{y}{\cosh x}}\]

C

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

↓

double code(double x, double y) {
	return (-sin(y) / -(y / cosh(x)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.1
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 0.1

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

3. Applied clear-num 0.2

   \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}\]
4. Using strategy rm

5. Applied div-inv 0.4

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

6. Applied associate-/r* 0.2

   \[\leadsto \cosh x \cdot \color{blue}{\frac{\frac{1}{y}}{\frac{1}{\sin y}}}\]
7. Using strategy rm

8. Applied clear-num 0.2

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

9. Applied associate-/r/ 0.3

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

10. Applied associate-*r* 0.3

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

11. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \frac{\sin y}{1}\]
12. Using strategy rm

13. Applied frac-2neg 0.3

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

14. Applied clear-num 0.3

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

15. Applied frac-times 0.2

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

16. Simplified 0.2

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

17. Simplified 0.2

    \[\leadsto \frac{-\sin y}{\color{blue}{-\frac{y}{\cosh x}}}\]

18. Final simplification 0.2

    \[\leadsto \frac{-\sin y}{-\frac{y}{\cosh x}}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

  (* (cosh x) (/ (sin y) y)))