Average Error: 0.0 → 0.1
Time: 6.3s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\left(\left(\left|\sqrt[3]{\frac{\sinh y}{y}}\right| \cdot \sin x\right) \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right) \cdot \sqrt{\frac{\sinh y}{y}}\]
\sin x \cdot \frac{\sinh y}{y}
↓
\left(\left(\left|\sqrt[3]{\frac{\sinh y}{y}}\right| \cdot \sin x\right) \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\right) \cdot \sqrt{\frac{\sinh y}{y}}
double f(double x, double y) {
        double r138784 = x;
        double r138785 = sin(r138784);
        double r138786 = y;
        double r138787 = sinh(r138786);
        double r138788 = r138787 / r138786;
        double r138789 = r138785 * r138788;
        return r138789;
}
↓
double f(double x, double y) {
        double r138790 = y;
        double r138791 = sinh(r138790);
        double r138792 = r138791 / r138790;
        double r138793 = cbrt(r138792);
        double r138794 = fabs(r138793);
        double r138795 = x;
        double r138796 = sin(r138795);
        double r138797 = r138794 * r138796;
        double r138798 = sqrt(r138793);
        double r138799 = r138797 * r138798;
        double r138800 = sqrt(r138792);
        double r138801 = r138799 * r138800;
        return r138801;
}

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. Applied associate-*r*0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

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)))