Average Error: 26.5 → 13.2
Time: 22.6s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
↓
\[\begin{array}{l} t_0 := 0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq 2.108913565740041 \cdot 10^{-21}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\ t_2 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\ t_3 := \sqrt{\frac{h \cdot t_0}{\ell}}\\ \left(\left(\left(\left|t_2\right| \cdot \sqrt{t_2}\right) \cdot \left(\left|t_1\right| \cdot \sqrt{t_1}\right)\right) \cdot \left(1 + t_3\right)\right) \cdot \left(1 - t_3\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 - t_0 \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
\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)
↓
\begin{array}{l}
t_0 := 0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\\
\mathbf{if}\;\ell \leq 2.108913565740041 \cdot 10^{-21}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\
t_2 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\
t_3 := \sqrt{\frac{h \cdot t_0}{\ell}}\\
\left(\left(\left(\left|t_2\right| \cdot \sqrt{t_2}\right) \cdot \left(\left|t_1\right| \cdot \sqrt{t_1}\right)\right) \cdot \left(1 + t_3\right)\right) \cdot \left(1 - t_3\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{d \cdot \left(1 - t_0 \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\


\end{array}

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
↓
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow (/ (* D M) (* d 2.0)) 2.0))))
   (if (<= l 2.108913565740041e-21)
     (let* ((t_1 (/ (cbrt d) (cbrt l)))
            (t_2 (/ (cbrt d) (cbrt h)))
            (t_3 (sqrt (/ (* h t_0) l))))
       (*
        (* (* (* (fabs t_2) (sqrt t_2)) (* (fabs t_1) (sqrt t_1))) (+ 1.0 t_3))
        (- 1.0 t_3)))
     (/ (* d (- 1.0 (* t_0 (/ h l)))) (* (sqrt h) (sqrt l))))))
double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}
↓
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * pow(((D * M) / (d * 2.0)), 2.0);
	double tmp;
	if (l <= 2.108913565740041e-21) {
		double t_1_1 = cbrt(d) / cbrt(l);
		double t_2_2 = cbrt(d) / cbrt(h);
		double t_3_3 = sqrt((h * t_0) / l);
		tmp = (((fabs(t_2_2) * sqrt(t_2_2)) * (fabs(t_1_1) * sqrt(t_1_1))) * (1.0 + t_3_3)) * (1.0 - t_3_3);
	} else {
		tmp = (d * (1.0 - (t_0 * (h / l)))) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if l < 2.1089135657400409e-21

    1. Initial program 26.8

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

    2. Simplified26.8

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]
    3. Using strategy rm

    4. Applied add-cube-cbrt_binary6427.1

      \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    5. Applied add-cube-cbrt_binary6427.2

      \[\leadsto \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    6. Applied times-frac_binary6427.2

      \[\leadsto \left(\sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    7. Applied sqrt-prod_binary6422.5

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    8. Simplified22.1

      \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
    9. Using strategy rm

    10. Applied add-cube-cbrt_binary6422.2

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    11. Applied add-cube-cbrt_binary6422.4

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    12. Applied times-frac_binary6422.4

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    13. Applied sqrt-prod_binary6418.3

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    14. Simplified18.1

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
    15. Using strategy rm

    16. Applied associate-*r/_binary6412.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

    17. Simplified12.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}}{\ell}\right) \]
    18. Using strategy rm

    19. Applied add-sqr-sqrt_binary6412.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}} \cdot \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}}\right) \]

    20. Applied *-un-lft-identity_binary6412.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\color{blue}{1 \cdot 1} - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}} \cdot \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}\right) \]

    21. Applied difference-of-squares_binary6412.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \color{blue}{\left(\left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}\right)\right)} \]

    22. Applied associate-*r*_binary6412.8

      \[\leadsto \color{blue}{\left(\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}\right)\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}\right)}{\ell}}\right)} \]

    if 2.1089135657400409e-21 < l

    1. Initial program 25.8

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

    2. Simplified25.8

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]
    3. Using strategy rm

    4. Applied sqrt-div_binary6422.1

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    5. Applied sqrt-div_binary6414.2

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    6. Applied frac-times_binary6414.2

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

    7. Applied associate-*l/_binary6414.0

      \[\leadsto \color{blue}{\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

    8. Simplified13.8

      \[\leadsto \frac{\color{blue}{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.108913565740041 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021202 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))