Average Error: 0.1 → 0.1
Time: 26.0s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\cosh x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin y}{y}\right)\right)\]
\cosh x \cdot \frac{\sin y}{y}
↓
\cosh x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin y}{y}\right)\right)
double f(double x, double y) {
        double r313400 = x;
        double r313401 = cosh(r313400);
        double r313402 = y;
        double r313403 = sin(r313402);
        double r313404 = r313403 / r313402;
        double r313405 = r313401 * r313404;
        return r313405;
}
↓
double f(double x, double y) {
        double r313406 = x;
        double r313407 = cosh(r313406);
        double r313408 = y;
        double r313409 = sin(r313408);
        double r313410 = r313409 / r313408;
        double r313411 = expm1(r313410);
        double r313412 = log1p(r313411);
        double r313413 = r313407 * r313412;
        return r313413;
}

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. Using strategy rm

  3. Applied log1p-expm1-u0.1

    \[\leadsto \cosh x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin y}{y}\right)\right)}\]

  4. Final simplification0.1

    \[\leadsto \cosh x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin y}{y}\right)\right)\]

Reproduce

herbie shell --seed 2019323 +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)))