Average Error: 14.0 → 0.1
Time: 1.1m
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\sin x}{x} \cdot \sinh y\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\sin x}{x} \cdot \sinh y
double f(double x, double y) {
        double r16500404 = x;
        double r16500405 = sin(r16500404);
        double r16500406 = y;
        double r16500407 = sinh(r16500406);
        double r16500408 = r16500405 * r16500407;
        double r16500409 = r16500408 / r16500404;
        return r16500409;
}
↓
double f(double x, double y) {
        double r16500410 = x;
        double r16500411 = sin(r16500410);
        double r16500412 = r16500411 / r16500410;
        double r16500413 = y;
        double r16500414 = sinh(r16500413);
        double r16500415 = r16500412 * r16500414;
        return r16500415;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original14.0
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.0

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

  3. Applied associate-/l*0.8

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

  4. Taylor expanded around inf 43.6

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019200 +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))