Average Error: 25.9 → 16.4
Time: 48.0s
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\mathsf{fma}\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right), \frac{-1}{2} \cdot \left(\frac{\frac{M}{\frac{d \cdot 2}{D}} \cdot h}{\ell} \cdot \frac{M}{\frac{d \cdot 2}{D}}\right), \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{d}{h}}\right)\]
\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)
↓
\mathsf{fma}\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right), \frac{-1}{2} \cdot \left(\frac{\frac{M}{\frac{d \cdot 2}{D}} \cdot h}{\ell} \cdot \frac{M}{\frac{d \cdot 2}{D}}\right), \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{d}{h}}\right)
double f(double d, double h, double l, double M, double D) {
        double r9698089 = d;
        double r9698090 = h;
        double r9698091 = r9698089 / r9698090;
        double r9698092 = 1.0;
        double r9698093 = 2.0;
        double r9698094 = r9698092 / r9698093;
        double r9698095 = pow(r9698091, r9698094);
        double r9698096 = l;
        double r9698097 = r9698089 / r9698096;
        double r9698098 = pow(r9698097, r9698094);
        double r9698099 = r9698095 * r9698098;
        double r9698100 = M;
        double r9698101 = D;
        double r9698102 = r9698100 * r9698101;
        double r9698103 = r9698093 * r9698089;
        double r9698104 = r9698102 / r9698103;
        double r9698105 = pow(r9698104, r9698093);
        double r9698106 = r9698094 * r9698105;
        double r9698107 = r9698090 / r9698096;
        double r9698108 = r9698106 * r9698107;
        double r9698109 = r9698092 - r9698108;
        double r9698110 = r9698099 * r9698109;
        return r9698110;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r9698111 = d;
        double r9698112 = cbrt(r9698111);
        double r9698113 = l;
        double r9698114 = cbrt(r9698113);
        double r9698115 = r9698112 / r9698114;
        double r9698116 = fabs(r9698115);
        double r9698117 = sqrt(r9698115);
        double r9698118 = r9698116 * r9698117;
        double r9698119 = h;
        double r9698120 = cbrt(r9698119);
        double r9698121 = r9698112 / r9698120;
        double r9698122 = fabs(r9698121);
        double r9698123 = sqrt(r9698121);
        double r9698124 = r9698122 * r9698123;
        double r9698125 = r9698118 * r9698124;
        double r9698126 = -0.5;
        double r9698127 = M;
        double r9698128 = 2.0;
        double r9698129 = r9698111 * r9698128;
        double r9698130 = D;
        double r9698131 = r9698129 / r9698130;
        double r9698132 = r9698127 / r9698131;
        double r9698133 = r9698132 * r9698119;
        double r9698134 = r9698133 / r9698113;
        double r9698135 = r9698134 * r9698132;
        double r9698136 = r9698126 * r9698135;
        double r9698137 = r9698111 / r9698119;
        double r9698138 = sqrt(r9698137);
        double r9698139 = r9698118 * r9698138;
        double r9698140 = fma(r9698125, r9698136, r9698139);
        return r9698140;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Initial program 25.9

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

  2. Simplified24.7

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

  4. Applied add-cube-cbrt24.9

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

  5. Applied add-cube-cbrt25.0

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

  6. Applied times-frac25.0

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

  7. Applied sqrt-prod22.9

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

  8. Simplified22.6

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

  10. Applied add-cube-cbrt22.6

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

  11. Applied add-cube-cbrt22.6

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

  12. Applied times-frac22.6

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

  13. Applied sqrt-prod20.6

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

  14. Simplified20.6

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

  16. Applied associate-*r/17.0

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

  18. Applied add-cube-cbrt17.0

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

  19. Applied add-cube-cbrt17.0

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

  20. Applied times-frac17.0

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

  21. Applied sqrt-prod16.4

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

  22. Simplified16.4

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

  23. Final simplification16.4

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

Reproduce

herbie shell --seed 2019163 +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)))))