Average Error: 14.7 → 0.1
Time: 4.4s
Precision: binary64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \left(2 \cdot \sinh y\right)}{2}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\frac{\sin x}{x} \cdot \left(2 \cdot \sinh y\right)}{2}
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) (((double) (((double) sin(x)) / x)) * ((double) (2.0 * ((double) sinh(y)))))) / 2.0));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original14.7
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.7

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

  3. Applied *-un-lft-identity14.7

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  7. Applied div-inv0.3

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

  9. Applied sinh-def44.0

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

  10. Applied associate-*l/44.0

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

  11. Applied associate-*r/44.0

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

  12. Simplified0.2

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

  14. Applied un-div-inv0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020174 
(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))