Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: 64

Math

\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]

↓

\[\left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + \left(\left(\frac{1}{3} \cdot \left(\frac{2}{3} \cdot \log y\right)\right) \cdot x + x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right)\right)\right) - y\right) - z\right) + \log t\]

C

double f(double x, double y, double z, double t) {
        double r121106 = x;
        double r121107 = y;
        double r121108 = log(r121107);
        double r121109 = r121106 * r121108;
        double r121110 = r121109 - r121107;
        double r121111 = z;
        double r121112 = r121110 - r121111;
        double r121113 = t;
        double r121114 = log(r121113);
        double r121115 = r121112 + r121114;
        return r121115;
}

↓

double f(double x, double y, double z, double t) {
        double r121116 = x;
        double r121117 = 2.0;
        double r121118 = y;
        double r121119 = cbrt(r121118);
        double r121120 = log(r121119);
        double r121121 = r121117 * r121120;
        double r121122 = r121116 * r121121;
        double r121123 = 0.0;
        double r121124 = 0.3333333333333333;
        double r121125 = 0.6666666666666666;
        double r121126 = log(r121118);
        double r121127 = r121125 * r121126;
        double r121128 = r121124 * r121127;
        double r121129 = r121128 * r121116;
        double r121130 = pow(r121119, r121124);
        double r121131 = log(r121130);
        double r121132 = r121116 * r121131;
        double r121133 = r121129 + r121132;
        double r121134 = r121123 + r121133;
        double r121135 = r121122 + r121134;
        double r121136 = r121135 - r121118;
        double r121137 = z;
        double r121138 = r121136 - r121137;
        double r121139 = t;
        double r121140 = log(r121139);
        double r121141 = r121138 + r121140;
        return r121141;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 0.1

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

   \[\leadsto \left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - y\right) - z\right) + \log t\]

4. Applied log-prod 0.1

   \[\leadsto \left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

5. Applied distribute-lft-in 0.1

   \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

6. Simplified 0.1

   \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
7. Using strategy rm

8. Applied *-un-lft-identity 0.1

   \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left(\sqrt[3]{\color{blue}{1 \cdot y}}\right)\right) - y\right) - z\right) + \log t\]

9. Applied cbrt-prod 0.1

   \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{y}\right)}\right) - y\right) - z\right) + \log t\]

10. Applied log-prod 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \color{blue}{\left(\log \left(\sqrt[3]{1}\right) + \log \left(\sqrt[3]{y}\right)\right)}\right) - y\right) - z\right) + \log t\]

11. Applied distribute-lft-in 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\left(x \cdot \log \left(\sqrt[3]{1}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) - y\right) - z\right) + \log t\]

12. Simplified 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\color{blue}{0} + x \cdot \log \left(\sqrt[3]{y}\right)\right)\right) - y\right) - z\right) + \log t\]

13. Simplified 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + \color{blue}{x \cdot \log \left({y}^{\frac{1}{3}}\right)}\right)\right) - y\right) - z\right) + \log t\]
14. Using strategy rm

15. Applied add-cube-cbrt 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + x \cdot \log \left({\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}}^{\frac{1}{3}}\right)\right)\right) - y\right) - z\right) + \log t\]

16. Applied unpow-prod-down 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + x \cdot \log \color{blue}{\left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{\frac{1}{3}} \cdot {\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)}\right)\right) - y\right) - z\right) + \log t\]

17. Applied log-prod 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + x \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{\frac{1}{3}}\right) + \log \left({\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right)}\right)\right) - y\right) - z\right) + \log t\]

18. Applied distribute-lft-in 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + \color{blue}{\left(x \cdot \log \left({\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{\frac{1}{3}}\right) + x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right)}\right)\right) - y\right) - z\right) + \log t\]

19. Simplified 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + \left(\color{blue}{\left(\frac{1}{3} \cdot \left(\frac{2}{3} \cdot \log y\right)\right) \cdot x} + x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right)\right)\right) - y\right) - z\right) + \log t\]

20. Final simplification 0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(0 + \left(\left(\frac{1}{3} \cdot \left(\frac{2}{3} \cdot \log y\right)\right) \cdot x + x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right)\right)\right) - y\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))