Average Error: 14.1 → 0.7
Time: 18.3s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\left(y \cdot \frac{\sin x}{x} + \frac{\frac{1}{6}}{\frac{x}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \sin x}}\right) + \frac{1}{120} \cdot \left(\frac{\sin x}{x} \cdot {y}^{5}\right)\]
\frac{\sin x \cdot \sinh y}{x}
↓
\left(y \cdot \frac{\sin x}{x} + \frac{\frac{1}{6}}{\frac{x}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \sin x}}\right) + \frac{1}{120} \cdot \left(\frac{\sin x}{x} \cdot {y}^{5}\right)
double f(double x, double y) {
        double r26214438 = x;
        double r26214439 = sin(r26214438);
        double r26214440 = y;
        double r26214441 = sinh(r26214440);
        double r26214442 = r26214439 * r26214441;
        double r26214443 = r26214442 / r26214438;
        return r26214443;
}
↓
double f(double x, double y) {
        double r26214444 = y;
        double r26214445 = x;
        double r26214446 = sin(r26214445);
        double r26214447 = r26214446 / r26214445;
        double r26214448 = r26214444 * r26214447;
        double r26214449 = 0.16666666666666666;
        double r26214450 = r26214444 * r26214444;
        double r26214451 = r26214444 * r26214450;
        double r26214452 = r26214451 * r26214446;
        double r26214453 = r26214445 / r26214452;
        double r26214454 = r26214449 / r26214453;
        double r26214455 = r26214448 + r26214454;
        double r26214456 = 0.008333333333333333;
        double r26214457 = 5.0;
        double r26214458 = pow(r26214444, r26214457);
        double r26214459 = r26214447 * r26214458;
        double r26214460 = r26214456 * r26214459;
        double r26214461 = r26214455 + r26214460;
        return r26214461;
}

Error

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Bits error versus y

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Results

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Target

Original14.1
Target0.2
Herbie0.7
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.1

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

  3. Applied *-un-lft-identity14.1

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  6. Taylor expanded around 0 0.7

    \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{6} \cdot {y}^{3} + \left(y + \frac{1}{120} \cdot {y}^{5}\right)}}{x}\]

  7. Simplified0.7

    \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{120} \cdot {y}^{5} + \left(\left(y \cdot y\right) \cdot \left(y \cdot \frac{1}{6}\right) + y\right)}}{x}\]

  8. Taylor expanded around inf 14.7

    \[\leadsto \color{blue}{\frac{1}{120} \cdot \frac{\sin x \cdot {y}^{5}}{x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot {y}^{3}}{x} + \frac{\sin x \cdot y}{x}\right)}\]

  9. Simplified0.7

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{6}}{\frac{x}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \sin x}} + \frac{\sin x}{x} \cdot y\right) + \left(\frac{\sin x}{x} \cdot {y}^{5}\right) \cdot \frac{1}{120}}\]

  10. Final simplification0.7

    \[\leadsto \left(y \cdot \frac{\sin x}{x} + \frac{\frac{1}{6}}{\frac{x}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \sin x}}\right) + \frac{1}{120} \cdot \left(\frac{\sin x}{x} \cdot {y}^{5}\right)\]

Reproduce

herbie shell --seed 2019158 
(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))