Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 13.9 → 0.7

Time: 16.3s

Precision: 64

Math

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

↓

\[\sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}}}\right), \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

C

double f(double x, double y) {
        double r351203 = x;
        double r351204 = sin(r351203);
        double r351205 = y;
        double r351206 = sinh(r351205);
        double r351207 = r351204 * r351206;
        double r351208 = r351207 / r351203;
        return r351208;
}

↓

double f(double x, double y) {
        double r351209 = x;
        double r351210 = sin(r351209);
        double r351211 = y;
        double r351212 = cbrt(r351211);
        double r351213 = r351212 * r351212;
        double r351214 = 5.0;
        double r351215 = pow(r351213, r351214);
        double r351216 = cbrt(r351209);
        double r351217 = r351216 * r351216;
        double r351218 = r351215 / r351217;
        double r351219 = cbrt(r351212);
        double r351220 = r351219 * r351219;
        double r351221 = pow(r351220, r351214);
        double r351222 = cbrt(r351216);
        double r351223 = r351222 * r351222;
        double r351224 = r351221 / r351223;
        double r351225 = pow(r351219, r351214);
        double r351226 = r351225 / r351222;
        double r351227 = r351224 * r351226;
        double r351228 = r351218 * r351227;
        double r351229 = 0.008333333333333333;
        double r351230 = 0.16666666666666666;
        double r351231 = 3.0;
        double r351232 = pow(r351211, r351231);
        double r351233 = r351232 / r351209;
        double r351234 = r351211 / r351209;
        double r351235 = fma(r351230, r351233, r351234);
        double r351236 = fma(r351228, r351229, r351235);
        double r351237 = r351210 * r351236;
        return r351237;
}

Error

Bits error versus x

Bits error versus y

Target

Original	13.9
Target	0.2
Herbie	0.7

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

Derivation

1. Initial program 13.9

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

3. Applied *-un-lft-identity 13.9

   \[\leadsto \frac{\sin x \cdot \sinh y}{\color{blue}{1 \cdot x}}\]

4. Applied times-frac 0.2

   \[\leadsto \color{blue}{\frac{\sin x}{1} \cdot \frac{\sinh y}{x}}\]

5. Simplified 0.2

   \[\leadsto \color{blue}{\sin x} \cdot \frac{\sinh y}{x}\]

6. Taylor expanded around 0 0.7

   \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{5}}{x} + \left(\frac{y}{x} + \frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)\right)}\]

7. Simplified 0.7

   \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{{y}^{5}}{x}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)}\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.7

   \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{y}^{5}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

10. Applied add-cube-cbrt 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}}^{5}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

11. Applied unpow-prod-down 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5} \cdot {\left(\sqrt[3]{y}\right)}^{5}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

12. Applied times-frac 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{{\left(\sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x}}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]
13. Using strategy rm

14. Applied add-cube-cbrt 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{{\left(\sqrt[3]{y}\right)}^{5}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

15. Applied add-cube-cbrt 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}^{5}}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

16. Applied unpow-prod-down 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}^{5} \cdot {\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{5}}}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

17. Applied times-frac 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}}}\right)}, \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

18. Final simplification 0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}}}\right), \frac{1}{120}, \mathsf{fma}\left(\frac{1}{6}, \frac{{y}^{3}}{x}, \frac{y}{x}\right)\right)\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))