Average Error: 26.2 → 13.5
Time: 15.5s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
↓
\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\ t_2 := \sqrt{t_1}\\ t_3 := \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_0\right) \cdot \frac{h}{\ell}\right) \leq 1.742499544712231 \cdot 10^{+217}:\\ \;\;\;\;\frac{\left(t_3 \cdot \left(\left|\sqrt[3]{d}\right| \cdot t_2\right)\right) \cdot \mathsf{fma}\left(t_0, \frac{h}{\ell} \cdot -0.5, 1\right)}{\left|\sqrt[3]{\ell}\right|}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \left|t_1\right|\right)\\ \end{array} \]
\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 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\
t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\
t_2 := \sqrt{t_1}\\
t_3 := \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\\
\mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_0\right) \cdot \frac{h}{\ell}\right) \leq 1.742499544712231 \cdot 10^{+217}:\\
\;\;\;\;\frac{\left(t_3 \cdot \left(\left|\sqrt[3]{d}\right| \cdot t_2\right)\right) \cdot \mathsf{fma}\left(t_0, \frac{h}{\ell} \cdot -0.5, 1\right)}{\left|\sqrt[3]{\ell}\right|}\\

\mathbf{else}:\\
\;\;\;\;t_3 \cdot \left(t_2 \cdot \left|t_1\right|\right)\\


\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 (pow (/ (* M D) (* d 2.0)) 2.0))
        (t_1 (/ (cbrt d) (cbrt l)))
        (t_2 (sqrt t_1))
        (t_3 (* (sqrt (/ 1.0 (* (cbrt h) (cbrt h)))) (sqrt (/ d (cbrt h))))))
   (if (<=
        (*
         (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
         (- 1.0 (* (* 0.5 t_0) (/ h l))))
        1.742499544712231e+217)
     (/
      (* (* t_3 (* (fabs (cbrt d)) t_2)) (fma t_0 (* (/ h l) -0.5) 1.0))
      (fabs (cbrt l)))
     (* t_3 (* t_2 (fabs t_1))))))
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 = pow(((M * D) / (d * 2.0)), 2.0);
	double t_1 = cbrt(d) / cbrt(l);
	double t_2 = sqrt(t_1);
	double t_3 = sqrt(1.0 / (cbrt(h) * cbrt(h))) * sqrt(d / cbrt(h));
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * t_0) * (h / l)))) <= 1.742499544712231e+217) {
		tmp = ((t_3 * (fabs(cbrt(d)) * t_2)) * fma(t_0, ((h / l) * -0.5), 1.0)) / fabs(cbrt(l));
	} else {
		tmp = t_3 * (t_2 * fabs(t_1));
	}
	return tmp;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.74249954471223101e217

    1. Initial program 12.9

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

    2. Simplified12.9

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

    3. Applied add-cube-cbrt_binary6413.2

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

    4. Applied add-cube-cbrt_binary6413.4

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

    5. Applied times-frac_binary6413.4

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

    6. Applied sqrt-prod_binary6410.0

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

    7. Simplified9.7

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

    8. Applied add-cube-cbrt_binary649.8

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

    9. Applied *-un-lft-identity_binary649.8

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

    10. Applied times-frac_binary649.8

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

    11. Applied sqrt-prod_binary648.7

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

    12. Applied fabs-div_binary648.7

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

    13. Applied associate-*l/_binary648.7

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

    14. Applied associate-*r/_binary648.7

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

    15. Applied associate-*l/_binary648.5

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

    if 1.74249954471223101e217 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

    1. Initial program 58.1

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

    2. Simplified58.1

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

    3. Applied add-cube-cbrt_binary6458.2

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

    4. Applied add-cube-cbrt_binary6458.2

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

    5. Applied times-frac_binary6458.2

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

    6. Applied sqrt-prod_binary6453.2

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

    7. Simplified53.2

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

    8. Applied add-cube-cbrt_binary6453.3

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

    9. Applied *-un-lft-identity_binary6453.3

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

    10. Applied times-frac_binary6453.3

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

    11. Applied sqrt-prod_binary6438.8

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

    12. Taylor expanded in M around 0 25.7

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 1.742499544712231 \cdot 10^{+217}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\left|\sqrt[3]{\ell}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022039 
(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)))))