Average Error: 14.2 → 0.1
Time: 21.6s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\sinh y}{\frac{1}{\frac{\sin x}{x}}}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\sinh y}{\frac{1}{\frac{\sin x}{x}}}
double f(double x, double y) {
        double r27817725 = x;
        double r27817726 = sin(r27817725);
        double r27817727 = y;
        double r27817728 = sinh(r27817727);
        double r27817729 = r27817726 * r27817728;
        double r27817730 = r27817729 / r27817725;
        return r27817730;
}
↓
double f(double x, double y) {
        double r27817731 = y;
        double r27817732 = sinh(r27817731);
        double r27817733 = 1.0;
        double r27817734 = x;
        double r27817735 = sin(r27817734);
        double r27817736 = r27817735 / r27817734;
        double r27817737 = r27817733 / r27817736;
        double r27817738 = r27817732 / r27817737;
        return r27817738;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 14.2

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

  2. Taylor expanded around inf 44.2

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

  3. Simplified0.1

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

  5. Applied clear-num0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 
(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))