Average Error: 14.4 → 0.7
Time: 20.3s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\left(\left(\frac{\sin x}{x} \cdot {y}^{5}\right) \cdot \frac{1}{120} + \frac{\sin x}{x} \cdot y\right) + \frac{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}{\frac{x}{\sin x}}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\left(\left(\frac{\sin x}{x} \cdot {y}^{5}\right) \cdot \frac{1}{120} + \frac{\sin x}{x} \cdot y\right) + \frac{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}{\frac{x}{\sin x}}
double f(double x, double y) {
        double r25091533 = x;
        double r25091534 = sin(r25091533);
        double r25091535 = y;
        double r25091536 = sinh(r25091535);
        double r25091537 = r25091534 * r25091536;
        double r25091538 = r25091537 / r25091533;
        return r25091538;
}
↓
double f(double x, double y) {
        double r25091539 = x;
        double r25091540 = sin(r25091539);
        double r25091541 = r25091540 / r25091539;
        double r25091542 = y;
        double r25091543 = 5.0;
        double r25091544 = pow(r25091542, r25091543);
        double r25091545 = r25091541 * r25091544;
        double r25091546 = 0.008333333333333333;
        double r25091547 = r25091545 * r25091546;
        double r25091548 = r25091541 * r25091542;
        double r25091549 = r25091547 + r25091548;
        double r25091550 = 0.16666666666666666;
        double r25091551 = r25091550 * r25091542;
        double r25091552 = r25091542 * r25091551;
        double r25091553 = r25091542 * r25091552;
        double r25091554 = r25091539 / r25091540;
        double r25091555 = r25091553 / r25091554;
        double r25091556 = r25091549 + r25091555;
        return r25091556;
}

Error

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Bits error versus y

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Results

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Target

Original14.4
Target0.2
Herbie0.7
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.4

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

  3. Applied *-un-lft-identity14.4

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

  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

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

  7. Simplified0.7

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

  8. Taylor expanded around inf 14.9

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

  9. Simplified0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019164 
(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))