Average Error: 26.7 → 10.0
Time: 5.2m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(1 - \left(h \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\left(1 - \left(h \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)
double f(double d, double h, double l, double M, double D) {
        double r46649049 = d;
        double r46649050 = h;
        double r46649051 = r46649049 / r46649050;
        double r46649052 = 1.0;
        double r46649053 = 2.0;
        double r46649054 = r46649052 / r46649053;
        double r46649055 = pow(r46649051, r46649054);
        double r46649056 = l;
        double r46649057 = r46649049 / r46649056;
        double r46649058 = pow(r46649057, r46649054);
        double r46649059 = r46649055 * r46649058;
        double r46649060 = M;
        double r46649061 = D;
        double r46649062 = r46649060 * r46649061;
        double r46649063 = r46649053 * r46649049;
        double r46649064 = r46649062 / r46649063;
        double r46649065 = pow(r46649064, r46649053);
        double r46649066 = r46649054 * r46649065;
        double r46649067 = r46649050 / r46649056;
        double r46649068 = r46649066 * r46649067;
        double r46649069 = r46649052 - r46649068;
        double r46649070 = r46649059 * r46649069;
        return r46649070;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r46649071 = 1.0;
        double r46649072 = h;
        double r46649073 = M;
        double r46649074 = D;
        double r46649075 = r46649073 * r46649074;
        double r46649076 = 2.0;
        double r46649077 = d;
        double r46649078 = r46649076 * r46649077;
        double r46649079 = r46649075 / r46649078;
        double r46649080 = cbrt(r46649076);
        double r46649081 = r46649079 / r46649080;
        double r46649082 = l;
        double r46649083 = cbrt(r46649082);
        double r46649084 = r46649081 / r46649083;
        double r46649085 = r46649072 * r46649084;
        double r46649086 = r46649083 * r46649080;
        double r46649087 = r46649079 / r46649086;
        double r46649088 = r46649087 / r46649086;
        double r46649089 = r46649085 * r46649088;
        double r46649090 = r46649071 - r46649089;
        double r46649091 = cbrt(r46649077);
        double r46649092 = r46649091 / r46649083;
        double r46649093 = 0.5;
        double r46649094 = pow(r46649092, r46649093);
        double r46649095 = fabs(r46649092);
        double r46649096 = r46649094 * r46649095;
        double r46649097 = cbrt(r46649072);
        double r46649098 = r46649091 / r46649097;
        double r46649099 = pow(r46649098, r46649093);
        double r46649100 = fabs(r46649098);
        double r46649101 = r46649099 * r46649100;
        double r46649102 = r46649096 * r46649101;
        double r46649103 = r46649090 * r46649102;
        return r46649103;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 26.7

    \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt26.9

    \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  4. Applied add-cube-cbrt27.0

    \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  5. Applied times-frac27.0

    \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  6. Applied unpow-prod-down21.9

    \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  7. Simplified21.2

    \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrt21.3

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  10. Applied add-cube-cbrt21.5

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  11. Applied times-frac21.4

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  12. Applied unpow-prod-down17.2

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  13. Simplified17.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
  14. Using strategy rm

  15. Applied pow117.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{h}{\ell}\right)}^{1}}\right)\]

  16. Applied pow117.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}^{1}} \cdot {\left(\frac{h}{\ell}\right)}^{1}\right)\]

  17. Applied pow-prod-down17.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{{\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}^{1}}\right)\]

  18. Simplified13.6

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}}{2}}{\ell} \cdot h\right)}}^{1}\right)\]
  19. Using strategy rm

  20. Applied add-cube-cbrt13.6

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{\frac{\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}}{2}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot h\right)}^{1}\right)\]

  21. Applied add-cube-cbrt13.6

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{\frac{\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot h\right)}^{1}\right)\]

  22. Applied times-frac13.6

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{\color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot h\right)}^{1}\right)\]

  23. Applied times-frac11.5

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\color{blue}{\left(\frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right)} \cdot h\right)}^{1}\right)\]

  24. Applied associate-*l*10.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}} \cdot h\right)\right)}}^{1}\right)\]

  25. Simplified10.0

    \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - {\left(\color{blue}{\frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}} \cdot \left(\frac{\frac{\frac{M \cdot D}{d \cdot 2}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}} \cdot h\right)\right)}^{1}\right)\]

  26. Final simplification10.0

    \[\leadsto \left(1 - \left(h \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))