Henrywood and Agarwal, Equation (9a)

Average Error: 14.6 → 10.0

Time: 15.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -2.793172886242329 \cdot 10^{+226}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{D \cdot \left(D \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 3.169537104913021 \cdot 10^{-112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}} \cdot \sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}}\right)\\ \end{array}\]

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 (<= (/ (* M D) (* 2.0 d)) -2.793172886242329e+226)
   (* w0 (* (sqrt (* (/ (* D (* D h)) (* d (* d l))) -0.25)) (- M)))
   (if (<= (/ (* M D) (* 2.0 d)) 3.169537104913021e-112)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* M D) (* 2.0 d)) 2.0)) l))))
     (*
      w0
      (*
       (sqrt
        (sqrt
         (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) (/ h l))))))
       (sqrt
        (sqrt
         (-
          1.0
          (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 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 (((M * D) / (2.0 * d)) <= -2.793172886242329e+226) {
		tmp = w0 * (sqrt(((D * (D * h)) / (d * (d * l))) * -0.25) * -M);
	} else if (((M * D) / (2.0 * d)) <= 3.169537104913021e-112) {
		tmp = w0 * sqrt(1.0 - ((h * pow(((M * D) / (2.0 * d)), 2.0)) / l));
	} else {
		tmp = w0 * (sqrt(sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * (h / l))))) * sqrt(sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * 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 (/.f64 (*.f64 M D) (*.f64 2 d)) < -2.7931728862423293e226

   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 unpow2_binary64_1166 64.0

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

   4. Applied associate-*l*_binary64_1042 61.4

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

   5. Taylor expanded around -inf 59.8

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

   6. Simplified 57.9

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

   if -2.7931728862423293e226 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 3.1695371049130211e-112

   1. Initial program 8.6

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

   3. Applied associate-*r/_binary64_1043 3.8

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

   if 3.1695371049130211e-112 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 25.1

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

   3. Applied unpow2_binary64_1166 25.1

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

   4. Applied associate-*l*_binary64_1042 21.4

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

   6. Applied add-sqr-sqrt_binary64_1123 21.5

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -2.793172886242329 \cdot 10^{+226}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{D \cdot \left(D \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 3.169537104913021 \cdot 10^{-112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}} \cdot \sqrt{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021025 
(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))))))