Average Error: 13.9 → 0.1
Time: 21.0s
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 r25370453 = x;
        double r25370454 = sin(r25370453);
        double r25370455 = y;
        double r25370456 = sinh(r25370455);
        double r25370457 = r25370454 * r25370456;
        double r25370458 = r25370457 / r25370453;
        return r25370458;
}
↓
double f(double x, double y) {
        double r25370459 = y;
        double r25370460 = sinh(r25370459);
        double r25370461 = x;
        double r25370462 = sin(r25370461);
        double r25370463 = r25370462 / r25370461;
        double r25370464 = r25370460 * r25370463;
        return r25370464;
}

Error

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Bits error versus y

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Results

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Target

Original13.9
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 13.9

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

  3. Applied associate-/l*0.7

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

  4. Taylor expanded around inf 43.5

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"

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

  (/ (* (sin x) (sinh y)) x))