Average Error: 0.1 → 0.2
Time: 23.2s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\sin y \cdot \left(e^{x} + e^{-x}\right)}{2 \cdot y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\sin y \cdot \left(e^{x} + e^{-x}\right)}{2 \cdot y}
double f(double x, double y) {
        double r372373 = x;
        double r372374 = cosh(r372373);
        double r372375 = y;
        double r372376 = sin(r372375);
        double r372377 = r372376 / r372375;
        double r372378 = r372374 * r372377;
        return r372378;
}
↓
double f(double x, double y) {
        double r372379 = y;
        double r372380 = sin(r372379);
        double r372381 = x;
        double r372382 = exp(r372381);
        double r372383 = -r372381;
        double r372384 = exp(r372383);
        double r372385 = r372382 + r372384;
        double r372386 = r372380 * r372385;
        double r372387 = 2.0;
        double r372388 = r372387 * r372379;
        double r372389 = r372386 / r372388;
        return r372389;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.1
Target0.1
Herbie0.2
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.1

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

  3. Applied cosh-def0.1

    \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{\sin y}{y}\]

  4. Applied frac-times0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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)))