Average Error: 0.1 → 0.1
Time: 15.2s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\frac{1}{2} \cdot \left(\left(e^{x} + e^{-x}\right) \cdot \sin y\right)}{y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\frac{1}{2} \cdot \left(\left(e^{x} + e^{-x}\right) \cdot \sin y\right)}{y}
double f(double x, double y) {
        double r384546 = x;
        double r384547 = cosh(r384546);
        double r384548 = y;
        double r384549 = sin(r384548);
        double r384550 = r384549 / r384548;
        double r384551 = r384547 * r384550;
        return r384551;
}
↓
double f(double x, double y) {
        double r384552 = 0.5;
        double r384553 = x;
        double r384554 = exp(r384553);
        double r384555 = -r384553;
        double r384556 = exp(r384555);
        double r384557 = r384554 + r384556;
        double r384558 = y;
        double r384559 = sin(r384558);
        double r384560 = r384557 * r384559;
        double r384561 = r384552 * r384560;
        double r384562 = r384561 / r384558;
        return r384562;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.1
Target0.1
Herbie0.1
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

  (* (cosh x) (/ (sin y) y)))