Average Error: 0.0 → 0.0
Time: 9.7s
Precision: 64
\[\cos x \cdot \frac{\sinh y}{y}\]
↓
\[\left(\left(\cos x \cdot \left(\sqrt{\frac{\sinh y}{y}} \cdot \left|\sqrt[3]{\frac{\sinh y}{y}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{\frac{\sinh y}{y}}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{\frac{\sinh y}{y}}}}\]
\cos x \cdot \frac{\sinh y}{y}
↓
\left(\left(\cos x \cdot \left(\sqrt{\frac{\sinh y}{y}} \cdot \left|\sqrt[3]{\frac{\sinh y}{y}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{\frac{\sinh y}{y}}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{\frac{\sinh y}{y}}}}
double f(double x, double y) {
        double r142856 = x;
        double r142857 = cos(r142856);
        double r142858 = y;
        double r142859 = sinh(r142858);
        double r142860 = r142859 / r142858;
        double r142861 = r142857 * r142860;
        return r142861;
}
↓
double f(double x, double y) {
        double r142862 = x;
        double r142863 = cos(r142862);
        double r142864 = y;
        double r142865 = sinh(r142864);
        double r142866 = r142865 / r142864;
        double r142867 = sqrt(r142866);
        double r142868 = cbrt(r142866);
        double r142869 = fabs(r142868);
        double r142870 = r142867 * r142869;
        double r142871 = r142863 * r142870;
        double r142872 = cbrt(r142867);
        double r142873 = sqrt(r142872);
        double r142874 = r142871 * r142873;
        double r142875 = r142874 * r142873;
        return r142875;
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-*r*0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied sqrt-prod0.0

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

  8. Applied associate-*r*0.0

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

  9. Simplified0.0

    \[\leadsto \color{blue}{\left(\cos x \cdot \left(\sqrt{\frac{\sinh y}{y}} \cdot \left|\sqrt[3]{\frac{\sinh y}{y}}\right|\right)\right)} \cdot \sqrt{\sqrt[3]{\frac{\sinh y}{y}}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.0

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

  12. Applied cbrt-prod0.0

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

  13. Applied sqrt-prod0.0

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

  14. Applied associate-*r*0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))