Henrywood and Agarwal, Equation (13)

Average Error: 48.3 → 35.4

Time: 53.2s

Precision: binary64

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

↓

\[\begin{array}{l} t_0 := \frac{c0}{D \cdot \left(w \cdot h\right)}\\ t_1 := d \cdot t_0\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M \leq -8.582941301532108 \cdot 10^{-100}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{d}{\frac{D}{t_1}}\right)\\ \mathbf{elif}\;M \leq -9.983596553371605 \cdot 10^{-128}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;M \leq -4.163638784959454 \cdot 10^{-173}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \left(t_1 \cdot \frac{d}{D}\right)\right)\\ \mathbf{elif}\;M \leq 5.141186143210871 \cdot 10^{-291}:\\ \;\;\;\;t_2 \cdot \sqrt{-{M}^{2}}\\ \mathbf{elif}\;M \leq 1.4373941293337057 \cdot 10^{-256}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{d}{\frac{\frac{D}{t_0}}{d}}\right)\\ \mathbf{elif}\;M \leq 4.474319272115561 \cdot 10^{-228}:\\ \;\;\;\;\log \left({\left(e^{\frac{c0}{w}}\right)}^{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \left(\left(d \cdot \frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}\right) \cdot \frac{d}{\frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* D (* w h)))) (t_1 (* d t_0)) (t_2 (/ c0 (* 2.0 w))))
   (if (<= M -8.582941301532108e-100)
     (* t_2 (* 2.0 (/ d (/ D t_1))))
     (if (<= M -9.983596553371605e-128)
       (*
        t_2
        (* 0.5 (/ (* (pow D 2.0) (* w (* h (pow M 2.0)))) (* c0 (pow d 2.0)))))
       (if (<= M -4.163638784959454e-173)
         (* t_2 (* 2.0 (* t_1 (/ d D))))
         (if (<= M 5.141186143210871e-291)
           (* t_2 (sqrt (- (pow M 2.0))))
           (if (<= M 1.4373941293337057e-256)
             (* t_2 (* 2.0 (/ d (/ (/ D t_0) d))))
             (if (<= M 4.474319272115561e-228)
               (log
                (pow (exp (/ c0 w)) (/ (* c0 (* d d)) (* (* w h) (* D D)))))
               (*
                t_2
                (*
                 2.0
                 (*
                  (* d (/ (* (cbrt c0) (cbrt c0)) D))
                  (/ d (/ D (/ (cbrt c0) (* w h)))))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (D * (w * h));
	double t_1 = d * t_0;
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (M <= -8.582941301532108e-100) {
		tmp = t_2 * (2.0 * (d / (D / t_1)));
	} else if (M <= -9.983596553371605e-128) {
		tmp = t_2 * (0.5 * ((pow(D, 2.0) * (w * (h * pow(M, 2.0)))) / (c0 * pow(d, 2.0))));
	} else if (M <= -4.163638784959454e-173) {
		tmp = t_2 * (2.0 * (t_1 * (d / D)));
	} else if (M <= 5.141186143210871e-291) {
		tmp = t_2 * sqrt(-pow(M, 2.0));
	} else if (M <= 1.4373941293337057e-256) {
		tmp = t_2 * (2.0 * (d / ((D / t_0) / d)));
	} else if (M <= 4.474319272115561e-228) {
		tmp = log(pow(exp(c0 / w), ((c0 * (d * d)) / ((w * h) * (D * D)))));
	} else {
		tmp = t_2 * (2.0 * ((d * ((cbrt(c0) * cbrt(c0)) / D)) * (d / (D / (cbrt(c0) / (w * h))))));
	}
	return tmp;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 7 regimes

2. if M < -8.5829413015321076e-100

   1. Initial program 50.5

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 42.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Applied unpow2_binary64 42.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right) \]

   4. Applied associate-*l*_binary64 40.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}\right) \]

   5. Applied associate-/l*_binary64 39.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{c0}}}\right) \]

   6. Simplified 39.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   7. Applied sqr-pow_binary64 39.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)} \cdot {d}^{\left(\frac{2}{2}\right)}}}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}\right) \]

   8. Applied associate-/l*_binary64 34.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{\left(\frac{2}{2}\right)}}{\frac{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}{{d}^{\left(\frac{2}{2}\right)}}}}\right) \]

   9. Simplified 33.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{D}{d \cdot \frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   if -8.5829413015321076e-100 < M < -9.9835965533716054e-128

   1. Initial program 45.9

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 44.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0}\right)} \]

   if -9.9835965533716054e-128 < M < -4.163638784959454e-173

   1. Initial program 47.5

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 47.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Applied unpow2_binary64 47.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right) \]

   4. Applied associate-*l*_binary64 44.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}\right) \]

   5. Applied associate-/l*_binary64 43.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{c0}}}\right) \]

   6. Simplified 43.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   7. Applied *-un-lft-identity_binary64 43.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{D}{\frac{\color{blue}{1 \cdot c0}}{D \cdot \left(w \cdot h\right)}}}\right) \]

   8. Applied times-frac_binary64 42.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{D}{\color{blue}{\frac{1}{D} \cdot \frac{c0}{w \cdot h}}}}\right) \]

   9. Applied *-un-lft-identity_binary64 42.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{\color{blue}{1 \cdot D}}{\frac{1}{D} \cdot \frac{c0}{w \cdot h}}}\right) \]

   10. Applied times-frac_binary64 42.8

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{1}{\frac{1}{D}} \cdot \frac{D}{\frac{c0}{w \cdot h}}}}\right) \]

   11. Applied add-sqr-sqrt_binary64 53.2

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)}}^{2}}{\frac{1}{\frac{1}{D}} \cdot \frac{D}{\frac{c0}{w \cdot h}}}\right) \]

   12. Applied unpow-prod-down_binary64 53.2

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{{\left(\sqrt{d}\right)}^{2} \cdot {\left(\sqrt{d}\right)}^{2}}}{\frac{1}{\frac{1}{D}} \cdot \frac{D}{\frac{c0}{w \cdot h}}}\right) \]

   13. Applied times-frac_binary64 51.8

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{d}\right)}^{2}}{\frac{1}{\frac{1}{D}}} \cdot \frac{{\left(\sqrt{d}\right)}^{2}}{\frac{D}{\frac{c0}{w \cdot h}}}\right)}\right) \]

   14. Simplified 51.8

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{d}{D}} \cdot \frac{{\left(\sqrt{d}\right)}^{2}}{\frac{D}{\frac{c0}{w \cdot h}}}\right)\right) \]

   15. Simplified 39.6

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(d \cdot \frac{c0}{D \cdot \left(w \cdot h\right)}\right)}\right)\right) \]

   if -4.163638784959454e-173 < M < 5.14118614321087072e-291

   1. Initial program 43.6

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0 37.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\sqrt{-{M}^{2}}} \]

   if 5.14118614321087072e-291 < M < 1.4373941293337057e-256

   1. Initial program 45.8

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 45.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Applied unpow2_binary64 45.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right) \]

   4. Applied associate-*l*_binary64 42.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}\right) \]

   5. Applied associate-/l*_binary64 42.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{c0}}}\right) \]

   6. Simplified 39.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   7. Applied unpow2_binary64 39.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{d \cdot d}}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}\right) \]

   8. Applied associate-/l*_binary64 33.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{d}{\frac{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}{d}}}\right) \]

   if 1.4373941293337057e-256 < M < 4.47431927211556072e-228

   1. Initial program 46.9

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 47.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Applied unpow2_binary64 47.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right) \]

   4. Applied associate-*l*_binary64 42.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}\right) \]

   5. Applied associate-/l*_binary64 42.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{c0}}}\right) \]

   6. Simplified 40.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   7. Applied add-log-exp_binary64 44.3

      \[\leadsto \color{blue}{\log \left(e^{\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}\right)}\right)} \]

   8. Simplified 39.3

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{c0}{w} \cdot 1}\right)}^{\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)}\right)} \]

   if 4.47431927211556072e-228 < M

   1. Initial program 49.2

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 43.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Applied unpow2_binary64 43.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right) \]

   4. Applied associate-*l*_binary64 40.8

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}\right) \]

   5. Applied associate-/l*_binary64 40.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{c0}}}\right) \]

   6. Simplified 39.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}}\right) \]

   7. Applied add-cube-cbrt_binary64 39.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{D}{\frac{\color{blue}{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \sqrt[3]{c0}}}{D \cdot \left(w \cdot h\right)}}}\right) \]

   8. Applied times-frac_binary64 39.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{D}{\color{blue}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D} \cdot \frac{\sqrt[3]{c0}}{w \cdot h}}}}\right) \]

   9. Applied *-un-lft-identity_binary64 39.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\frac{\color{blue}{1 \cdot D}}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D} \cdot \frac{\sqrt[3]{c0}}{w \cdot h}}}\right) \]

   10. Applied times-frac_binary64 39.4

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2}}{\color{blue}{\frac{1}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}} \cdot \frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}}\right) \]

   11. Applied add-sqr-sqrt_binary64 51.6

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)}}^{2}}{\frac{1}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}} \cdot \frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right) \]

   12. Applied unpow-prod-down_binary64 51.6

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{{\left(\sqrt{d}\right)}^{2} \cdot {\left(\sqrt{d}\right)}^{2}}}{\frac{1}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}} \cdot \frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right) \]

   13. Applied times-frac_binary64 49.1

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{d}\right)}^{2}}{\frac{1}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}}} \cdot \frac{{\left(\sqrt{d}\right)}^{2}}{\frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right)}\right) \]

   14. Simplified 49.1

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\left(d \cdot \frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}\right)} \cdot \frac{{\left(\sqrt{d}\right)}^{2}}{\frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right)\right) \]

   15. Simplified 34.4

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot \frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}\right) \cdot \color{blue}{\frac{d}{\frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}}\right)\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 35.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -8.582941301532108 \cdot 10^{-100}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d}{\frac{D}{d \cdot \frac{c0}{D \cdot \left(w \cdot h\right)}}}\right)\\ \mathbf{elif}\;M \leq -9.983596553371605 \cdot 10^{-128}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;M \leq -4.163638784959454 \cdot 10^{-173}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot \frac{c0}{D \cdot \left(w \cdot h\right)}\right) \cdot \frac{d}{D}\right)\right)\\ \mathbf{elif}\;M \leq 5.141186143210871 \cdot 10^{-291}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt{-{M}^{2}}\\ \mathbf{elif}\;M \leq 1.4373941293337057 \cdot 10^{-256}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d}{\frac{\frac{D}{\frac{c0}{D \cdot \left(w \cdot h\right)}}}{d}}\right)\\ \mathbf{elif}\;M \leq 4.474319272115561 \cdot 10^{-228}:\\ \;\;\;\;\log \left({\left(e^{\frac{c0}{w}}\right)}^{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot \frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{D}\right) \cdot \frac{d}{\frac{D}{\frac{\sqrt[3]{c0}}{w \cdot h}}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))