Average Error: 0.1 → 0.2
Time: 4.2s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\frac{\sin y \cdot 2}{\frac{y}{\cosh x}}}{2}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\frac{\sin y \cdot 2}{\frac{y}{\cosh x}}}{2}
double code(double x, double y) {
	return (cosh(x) * (sin(y) / y));
}
↓
double code(double x, double y) {
	return (((sin(y) * 2.0) / (y / cosh(x))) / 2.0);
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.1
Target0.1
Herbie0.2
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.1

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

  3. Applied clear-num0.2

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

  5. Applied div-inv0.4

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

  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 cosh-def0.2

    \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}\]

  9. Applied associate-*l/0.2

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

  10. Simplified0.3

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

  12. Applied clear-num0.3

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

  13. Applied un-div-inv0.3

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

  14. Applied associate-*l/0.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

    \[\leadsto \frac{\frac{\sin y \cdot 2}{\frac{y}{\cosh x}}}{2}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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)))