Average Error: 0.0 → 0.7
Time: 6.0s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\sin x \cdot e^{\sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}\]
\sin x \cdot \frac{\sinh y}{y}
↓
\sin x \cdot e^{\sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}
double f(double x, double y) {
        double r202763 = x;
        double r202764 = sin(r202763);
        double r202765 = y;
        double r202766 = sinh(r202765);
        double r202767 = r202766 / r202765;
        double r202768 = r202764 * r202767;
        return r202768;
}
↓
double f(double x, double y) {
        double r202769 = x;
        double r202770 = sin(r202769);
        double r202771 = 0.16666666666666666;
        double r202772 = y;
        double r202773 = 2.0;
        double r202774 = pow(r202772, r202773);
        double r202775 = 0.008333333333333333;
        double r202776 = 4.0;
        double r202777 = pow(r202772, r202776);
        double r202778 = 1.0;
        double r202779 = fma(r202775, r202777, r202778);
        double r202780 = fma(r202771, r202774, r202779);
        double r202781 = log(r202780);
        double r202782 = 3.0;
        double r202783 = pow(r202781, r202782);
        double r202784 = sqrt(r202783);
        double r202785 = cbrt(r202784);
        double r202786 = r202785 * r202785;
        double r202787 = exp(r202786);
        double r202788 = r202770 * r202787;
        return r202788;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\sin x \cdot \frac{\sinh y}{y}\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \left(\frac{1}{120} \cdot {y}^{4} + 1\right)\right)}\]

  3. Simplified0.7

    \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)}\]
  4. Using strategy rm

  5. Applied add-exp-log0.7

    \[\leadsto \sin x \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube0.7

    \[\leadsto \sin x \cdot e^{\color{blue}{\sqrt[3]{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)}}}\]

  8. Simplified0.7

    \[\leadsto \sin x \cdot e^{\sqrt[3]{\color{blue}{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.7

    \[\leadsto \sin x \cdot e^{\sqrt[3]{\color{blue}{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}} \cdot \sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}}\]

  11. Applied cbrt-prod0.7

    \[\leadsto \sin x \cdot e^{\color{blue}{\sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}}\]

  12. Final simplification0.7

    \[\leadsto \sin x \cdot e^{\sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left(\log \left(\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\right)\right)}^{3}}}}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))