Henrywood and Agarwal, Equation (9a)

Average Error: 14.1 → 9.6

Time: 14.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\frac{h \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.25}\right)\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.5508693631259018 \cdot 10^{+289}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

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 (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      (- INFINITY))
   (* w0 (* D (sqrt (* (/ (* h (* M M)) (* l (* d d))) -0.25))))
   (if (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
        1.5508693631259018e+289)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M (/ (* 2.0 d) D)) 2.0)))))
     (if (<=
          (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
          INFINITY)
       (-
        (* (/ w0 (* M (sqrt (* -0.25 (* (/ h l) (pow (/ D d) 2.0)))))) -0.5)
        (* M (* w0 (sqrt (* -0.25 (* (/ h l) (pow (/ D d) 2.0)))))))
       (*
        w0
        (sqrt
         (-
          1.0
          (*
           (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) h))
           (/ 1.0 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 ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= -((double) INFINITY)) {
		tmp = w0 * (D * sqrt(((h * (M * M)) / (l * (d * d))) * -0.25));
	} else if ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.5508693631259018e+289) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow((M / ((2.0 * d) / D)), 2.0)));
	} else if ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= ((double) INFINITY)) {
		tmp = ((w0 / (M * sqrt(-0.25 * ((h / l) * pow((D / d), 2.0))))) * -0.5) - (M * (w0 * sqrt(-0.25 * ((h / l) * pow((D / d), 2.0)))));
	} else {
		tmp = w0 * sqrt(1.0 - ((((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * h)) * (1.0 / 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 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 57.4

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

   3. Simplified 57.4

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

   if -inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 1.5508693631259018e289

   1. Initial program 0.2

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

   3. Applied associate-/l*_binary64 0.8

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

   if 1.5508693631259018e289 < (*.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 55.1

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

   2. Taylor expanded around -inf 56.5

      \[\leadsto \color{blue}{-\left(0.5 \cdot \frac{w0}{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}} + M \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot w0\right)\right)}\]

   3. Simplified 50.5

      \[\leadsto \color{blue}{\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot w0\right)}\]

   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 div-inv_binary64 64.0

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

   4. Applied associate-*r*_binary64 25.5

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

   5. Simplified 25.5

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

   7. Applied unpow2_binary64 25.5

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

   8. Applied associate-*r*_binary64 14.8

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\frac{h \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.25}\right)\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.5508693631259018 \cdot 10^{+289}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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))))))