Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 8.2

Time: 15.7s

Precision: binary64

\[[M, D]=\mathsf{sort}([M, D])\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ t_1 := w0 \cdot \sqrt{1 - {t_0}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;t_1 \leq 4.147320765878996 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\\ t_3 := \sqrt[3]{t_0}\\ w0 \cdot \sqrt{1 - t_2 \cdot \left(\left(\left(t_3 \cdot \left(t_3 \cdot t_3\right)\right) \cdot t_2\right) \cdot \left(t_0 \cdot t_2\right)\right)} \end{array}\\ \end{array} \]

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
 (let* ((t_0 (/ (* M D) (* 2.0 d)))
        (t_1 (* w0 (sqrt (- 1.0 (* (pow t_0 2.0) (/ h l)))))))
   (if (<= t_1 (- INFINITY))
     (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M)))
     (if (<= t_1 4.147320765878996e+306)
       (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
       (if (<= t_1 INFINITY)
         (* w0 (* D (sqrt (* -0.25 (* (/ h l) (pow (/ M d) 2.0))))))
         (let* ((t_2 (/ (cbrt h) (cbrt l))) (t_3 (cbrt t_0)))
           (*
            w0
            (sqrt
             (-
              1.0
              (* t_2 (* (* (* t_3 (* t_3 t_3)) t_2) (* t_0 t_2))))))))))))

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 t_0 = (M * D) / (2.0 * d);
	double t_1 = w0 * sqrt(1.0 - (pow(t_0, 2.0) * (h / l)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = w0 * (sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -M);
	} else if (t_1 <= 4.147320765878996e+306) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = w0 * (D * sqrt(-0.25 * ((h / l) * pow((M / d), 2.0))));
	} else {
		double t_2 = cbrt(h) / cbrt(l);
		double t_3 = cbrt(t_0);
		tmp = w0 * sqrt(1.0 - (t_2 * (((t_3 * (t_3 * t_3)) * t_2) * (t_0 * t_2))));
	}
	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 4 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 58.4

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell \cdot {d}^{2}}} \cdot M\right)\right)} \]

   3. Simplified 49.4

      \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)} \]

   if -inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 4.1473207658789957e306

   1. Initial program 0.1

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

   3. Applied times-frac_binary64 0.8

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

   if 4.1473207658789957e306 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 62.8

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

   3. Applied add-cube-cbrt_binary64 62.8

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

   4. Applied add-cube-cbrt_binary64 62.8

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

   5. Applied times-frac_binary64 62.8

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

   6. Applied associate-*r*_binary64 59.0

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

   7. Simplified 59.0

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

   8. Taylor expanded around inf 57.8

      \[\leadsto w0 \cdot \color{blue}{\left(D \cdot \sqrt{-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}}\right)} \]

   9. Simplified 47.2

      \[\leadsto w0 \cdot \color{blue}{\left(D \cdot \sqrt{\left({\left(\frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.25}\right)} \]

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 64.0

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

   3. Applied add-cube-cbrt_binary64 64.0

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

   4. Applied add-cube-cbrt_binary64 64.0

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

   5. Applied times-frac_binary64 64.0

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

   6. Applied associate-*r*_binary64 32.9

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

   7. Simplified 32.9

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

   9. Applied times-frac_binary64 32.9

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

   10. Applied add-sqr-sqrt_binary64 38.1

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

   11. Applied unpow-prod-down_binary64 38.1

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

   12. Applied unswap-sqr_binary64 24.8

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

   13. Simplified 24.8

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

   14. Simplified 12.1

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

   16. Applied add-cube-cbrt_binary64 12.1

       \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\left(\left(\sqrt[3]{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt[3]{\frac{D \cdot M}{d \cdot 2}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{d \cdot 2}}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4.147320765878996 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))))