Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 12.6

Time: 16.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\ t_1 := \sqrt{t_0}\\ t_2 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_3 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\ t_4 := \left|t_3\right| \cdot \sqrt{t_3}\\ t_5 := \left|t_0\right|\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_2\right) \cdot \frac{h}{\ell}\right) \leq 2.7780224486901018 \cdot 10^{+188}:\\ \;\;\;\;t_5 \cdot \left(t_1 \cdot \left(t_4 \cdot \mathsf{fma}\left(t_2, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(t_5 \cdot t_1\right)\\ \end{array} \]

FPCore

(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 (/ (cbrt d) (cbrt h)))
        (t_1 (sqrt t_0))
        (t_2 (pow (/ (* M D) (* d 2.0)) 2.0))
        (t_3 (/ (cbrt d) (cbrt l)))
        (t_4 (* (fabs t_3) (sqrt t_3)))
        (t_5 (fabs t_0)))
   (if (<=
        (*
         (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
         (- 1.0 (* (* 0.5 t_2) (/ h l))))
        2.7780224486901018e+188)
     (* t_5 (* t_1 (* t_4 (fma t_2 (* (/ h l) -0.5) 1.0))))
     (* t_4 (* t_5 t_1)))))

C

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 = cbrt(d) / cbrt(h);
	double t_1 = sqrt(t_0);
	double t_2 = pow(((M * D) / (d * 2.0)), 2.0);
	double t_3 = cbrt(d) / cbrt(l);
	double t_4 = fabs(t_3) * sqrt(t_3);
	double t_5 = fabs(t_0);
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * t_2) * (h / l)))) <= 2.7780224486901018e+188) {
		tmp = t_5 * (t_1 * (t_4 * fma(t_2, ((h / l) * -0.5), 1.0)));
	} else {
		tmp = t_4 * (t_5 * 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)))) < 2.77802244869010183e188

   1. Initial program 13.5

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

   2. Simplified 13.5

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

   3. Applied add-cube-cbrt_binary64 13.8

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

   4. Applied add-cube-cbrt_binary64 14.0

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

   5. Applied times-frac_binary64 14.0

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

   6. Applied sqrt-prod_binary64 12.8

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

   7. Simplified 12.8

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

   8. Applied add-cube-cbrt_binary64 12.9

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

   9. Applied add-cube-cbrt_binary64 13.1

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

   10. Applied times-frac_binary64 13.1

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

   11. Applied sqrt-prod_binary64 9.5

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

   12. Simplified 9.2

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

   13. Applied associate-*l*_binary64 8.4

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

   14. Applied associate-*l*_binary64 8.4

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

   if 2.77802244869010183e188 < (*.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 54.8

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

   2. Simplified 54.8

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

   3. Applied add-cube-cbrt_binary64 54.9

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

   4. Applied add-cube-cbrt_binary64 54.9

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

   5. Applied times-frac_binary64 54.9

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

   6. Applied sqrt-prod_binary64 40.4

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

   7. Simplified 38.5

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

   8. Applied add-cube-cbrt_binary64 38.5

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

   9. Applied add-cube-cbrt_binary64 38.6

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

   10. Applied times-frac_binary64 38.6

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

   11. Applied sqrt-prod_binary64 33.8

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

   12. Simplified 33.8

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

   13. Applied associate-*l*_binary64 33.8

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

   14. Taylor expanded in M around 0 21.8

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.7780224486901018 \cdot 10^{+188}:\\ \;\;\;\;\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))))