Average Error: 0.1 → 0.1
Time: 17.2s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}
double f(double x, double y) {
        double r787201 = x;
        double r787202 = cosh(r787201);
        double r787203 = y;
        double r787204 = sin(r787203);
        double r787205 = r787204 / r787203;
        double r787206 = r787202 * r787205;
        return r787206;
}
↓
double f(double x, double y) {
        double r787207 = 0.5;
        double r787208 = y;
        double r787209 = sin(r787208);
        double r787210 = r787207 * r787209;
        double r787211 = x;
        double r787212 = exp(r787211);
        double r787213 = -r787211;
        double r787214 = exp(r787213);
        double r787215 = r787212 + r787214;
        double r787216 = r787210 * r787215;
        double r787217 = r787216 / r787208;
        return r787217;
}

Error

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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

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 
(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)))