Average Error: 13.9 → 0.7
Time: 16.8s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\sin x \cdot \left(\frac{1}{120} \cdot \left(\frac{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{5}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{5}}{\sqrt[3]{\sqrt[3]{x}}}\right)\right) + \left(\frac{y}{x} + \frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)\right)\]
\frac{\sin x \cdot \sinh y}{x}
↓
\sin x \cdot \left(\frac{1}{120} \cdot \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)\right) + \left(\frac{y}{x} + \frac{1}{6} \cdot \frac{{y}^{3}}{x}\right)\right)
double f(double x, double y) {
        double r372127 = x;
        double r372128 = sin(r372127);
        double r372129 = y;
        double r372130 = sinh(r372129);
        double r372131 = r372128 * r372130;
        double r372132 = r372131 / r372127;
        return r372132;
}
↓
double f(double x, double y) {
        double r372133 = x;
        double r372134 = sin(r372133);
        double r372135 = 0.008333333333333333;
        double r372136 = y;
        double r372137 = cbrt(r372136);
        double r372138 = r372137 * r372137;
        double r372139 = 5.0;
        double r372140 = pow(r372138, r372139);
        double r372141 = cbrt(r372133);
        double r372142 = r372141 * r372141;
        double r372143 = r372140 / r372142;
        double r372144 = cbrt(r372137);
        double r372145 = r372144 * r372144;
        double r372146 = pow(r372145, r372139);
        double r372147 = cbrt(r372141);
        double r372148 = r372147 * r372147;
        double r372149 = r372146 / r372148;
        double r372150 = pow(r372144, r372139);
        double r372151 = r372150 / r372147;
        double r372152 = r372149 * r372151;
        double r372153 = r372143 * r372152;
        double r372154 = r372135 * r372153;
        double r372155 = r372136 / r372133;
        double r372156 = 0.16666666666666666;
        double r372157 = 3.0;
        double r372158 = pow(r372136, r372157);
        double r372159 = r372158 / r372133;
        double r372160 = r372156 * r372159;
        double r372161 = r372155 + r372160;
        double r372162 = r372154 + r372161;
        double r372163 = r372134 * r372162;
        return r372163;
}

Error

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Results

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Target

Original13.9
Target0.2
Herbie0.7
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 13.9

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

  3. Applied *-un-lft-identity13.9

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  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. Using strategy rm

  8. Applied add-cube-cbrt0.7

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

  9. Applied add-cube-cbrt0.7

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

  10. Applied unpow-prod-down0.7

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

  11. Applied times-frac0.7

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

  13. Applied add-cube-cbrt0.7

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

  14. Applied add-cube-cbrt0.7

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

  15. Applied unpow-prod-down0.7

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

  16. Applied times-frac0.7

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

  17. Final simplification0.7

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

Reproduce

herbie shell --seed 2019235 
(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))