Average Error: 13.8 → 0.1
Time: 4.9s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\frac{\sin x}{\frac{x}{2}} \cdot \sinh y}{2}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\frac{\sin x}{\frac{x}{2}} \cdot \sinh y}{2}
double f(double x, double y) {
        double r511031 = x;
        double r511032 = sin(r511031);
        double r511033 = y;
        double r511034 = sinh(r511033);
        double r511035 = r511032 * r511034;
        double r511036 = r511035 / r511031;
        return r511036;
}
↓
double f(double x, double y) {
        double r511037 = x;
        double r511038 = sin(r511037);
        double r511039 = 2.0;
        double r511040 = r511037 / r511039;
        double r511041 = r511038 / r511040;
        double r511042 = y;
        double r511043 = sinh(r511042);
        double r511044 = r511041 * r511043;
        double r511045 = r511044 / r511039;
        return r511045;
}

Error

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Bits error versus y

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Results

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Target

Original13.8
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 13.8

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

  3. Applied associate-/l*0.8

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

  5. Applied sinh-def43.1

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

  6. Applied associate-/r/43.1

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

  7. Applied associate-/r*43.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020065 +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))