Henrywood and Agarwal, Equation (9a)

Average Error: 14.0 → 8.7

Time: 24.9s

Precision: binary64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5.0984987494098346 \cdot 10^{-160}:\\ \;\;\;\;w0 \cdot \sqrt[3]{{\left(\sqrt{1 - \frac{\frac{\frac{D}{d}}{\frac{2}{M}} \cdot \left(h \cdot \frac{\frac{D}{d}}{\frac{2}{M}}\right)}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 7.944018704189104 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right)}\\ \end{array}\]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 5.0984987494098346e-160)
   (*
    w0
    (cbrt
     (pow
      (sqrt
       (- 1.0 (/ (* (/ (/ D d) (/ 2.0 M)) (* h (/ (/ D d) (/ 2.0 M)))) l)))
      3.0)))
   (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 7.944018704189104e+257)
     (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
     (*
      w0
      (sqrt
       (- 1.0 (* (* (/ D d) (/ M 2.0)) (* (/ M 2.0) (* (/ D d) (/ h l))))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 5.0984987494098346e-160) {
		tmp = w0 * cbrt(pow(sqrt(1.0 - ((((D / d) / (2.0 / M)) * (h * ((D / d) / (2.0 / M)))) / l)), 3.0));
	} else if (pow(((M * D) / (2.0 * d)), 2.0) <= 7.944018704189104e+257) {
		tmp = w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)));
	} else {
		tmp = w0 * sqrt(1.0 - (((D / d) * (M / 2.0)) * ((M / 2.0) * ((D / d) * (h / l)))));
	}
	return tmp;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0984987494098346e-160

   1. Initial program 6.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 6.4

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity_binary64_1101 6.4

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\color{blue}{1 \cdot \ell}}}\]

   5. Applied add-cube-cbrt_binary64_1136 6.4

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}}\]

   6. Applied times-frac_binary64_1107 6.4

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}}\]

   7. Applied associate-*r*_binary64_1041 2.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}}\]

   8. Simplified 2.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]
   9. Using strategy rm

   10. Applied associate-*r*_binary64_1041 2.5

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]

   11. Simplified 2.5

       \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left({\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \sqrt[3]{h}\right)} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\]
   12. Using strategy rm

   13. Applied add-cbrt-cube_binary64_1137 2.6

       \[\leadsto w0 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{1 - \left(\left({\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}} \cdot \sqrt{1 - \left(\left({\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\right) \cdot \sqrt{1 - \left(\left({\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}}}\]

   14. Simplified 1.2

       \[\leadsto w0 \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{1 - \frac{h \cdot {\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}}{\ell}}\right)}^{3}}}\]
   15. Using strategy rm

   16. Applied unpow2_binary64_1166 1.2

       \[\leadsto w0 \cdot \sqrt[3]{{\left(\sqrt{1 - \frac{h \cdot \color{blue}{\left(\frac{\frac{D}{d}}{\frac{2}{M}} \cdot \frac{\frac{D}{d}}{\frac{2}{M}}\right)}}{\ell}}\right)}^{3}}\]

   17. Applied associate-*r*_binary64_1041 0.6

       \[\leadsto w0 \cdot \sqrt[3]{{\left(\sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{\frac{D}{d}}{\frac{2}{M}}\right) \cdot \frac{\frac{D}{d}}{\frac{2}{M}}}}{\ell}}\right)}^{3}}\]

   if 5.0984987494098346e-160 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 7.94401870418910368e257

   1. Initial program 6.8

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   if 7.94401870418910368e257 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 59.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 56.5

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}\]
   3. Using strategy rm

   4. Applied unpow2_binary64_1166 56.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}}\]

   5. Applied associate-*l*_binary64_1042 45.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right)}}\]

   6. Simplified 46.1

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5.0984987494098346 \cdot 10^{-160}:\\ \;\;\;\;w0 \cdot \sqrt[3]{{\left(\sqrt{1 - \frac{\frac{\frac{D}{d}}{\frac{2}{M}} \cdot \left(h \cdot \frac{\frac{D}{d}}{\frac{2}{M}}\right)}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 7.944018704189104 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))