Average Error: 14.6 → 0.1
Time: 22.8s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\sinh y \cdot \frac{\sin x}{x}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\sinh y \cdot \frac{\sin x}{x}
double f(double x, double y) {
        double r446321 = x;
        double r446322 = sin(r446321);
        double r446323 = y;
        double r446324 = sinh(r446323);
        double r446325 = r446322 * r446324;
        double r446326 = r446325 / r446321;
        return r446326;
}
↓
double f(double x, double y) {
        double r446327 = y;
        double r446328 = sinh(r446327);
        double r446329 = x;
        double r446330 = sin(r446329);
        double r446331 = r446330 / r446329;
        double r446332 = r446328 * r446331;
        return r446332;
}

Error

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Results

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Target

Original14.6
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.6

    \[\frac{\sin x \cdot \sinh y}{x}\]
  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 *-un-lft-identity0.8

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

  6. Applied *-un-lft-identity0.8

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

  7. Applied times-frac0.8

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

  8. Applied *-un-lft-identity0.8

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

  9. Applied times-frac0.8

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

  10. Simplified0.8

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))