Average Error: 0.0 → 0.1
Time: 6.2s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\left(\sin x \cdot \left(\left|\sqrt[3]{\frac{\sinh y}{y}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right)\right) \cdot \sqrt{\frac{\sinh y}{y}}\]
\sin x \cdot \frac{\sinh y}{y}
↓
\left(\sin x \cdot \left(\left|\sqrt[3]{\frac{\sinh y}{y}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right)\right) \cdot \sqrt{\frac{\sinh y}{y}}
double f(double x, double y) {
        double r137396 = x;
        double r137397 = sin(r137396);
        double r137398 = y;
        double r137399 = sinh(r137398);
        double r137400 = r137399 / r137398;
        double r137401 = r137397 * r137400;
        return r137401;
}
↓
double f(double x, double y) {
        double r137402 = x;
        double r137403 = sin(r137402);
        double r137404 = y;
        double r137405 = sinh(r137404);
        double r137406 = r137405 / r137404;
        double r137407 = cbrt(r137406);
        double r137408 = fabs(r137407);
        double r137409 = sqrt(r137407);
        double r137410 = r137408 * r137409;
        double r137411 = r137403 * r137410;
        double r137412 = sqrt(r137406);
        double r137413 = r137411 * r137412;
        return r137413;
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \sin x \cdot \color{blue}{\left(\sqrt{\frac{\sinh y}{y}} \cdot \sqrt{\frac{\sinh y}{y}}\right)}\]

  4. Applied associate-*r*0.0

    \[\leadsto \color{blue}{\left(\sin x \cdot \sqrt{\frac{\sinh y}{y}}\right) \cdot \sqrt{\frac{\sinh y}{y}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.1

    \[\leadsto \left(\sin x \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{\sinh y}{y}} \cdot \sqrt[3]{\frac{\sinh y}{y}}\right) \cdot \sqrt[3]{\frac{\sinh y}{y}}}}\right) \cdot \sqrt{\frac{\sinh y}{y}}\]

  7. Applied sqrt-prod0.1

    \[\leadsto \left(\sin x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\frac{\sinh y}{y}} \cdot \sqrt[3]{\frac{\sinh y}{y}}} \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right)}\right) \cdot \sqrt{\frac{\sinh y}{y}}\]

  8. Simplified0.1

    \[\leadsto \left(\sin x \cdot \left(\color{blue}{\left|\sqrt[3]{\frac{\sinh y}{y}}\right|} \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right)\right) \cdot \sqrt{\frac{\sinh y}{y}}\]

  9. Final simplification0.1

    \[\leadsto \left(\sin x \cdot \left(\left|\sqrt[3]{\frac{\sinh y}{y}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right)\right) \cdot \sqrt{\frac{\sinh y}{y}}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))