Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 26.1

Time: 22.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -6.153227891404315 \cdot 10^{+162}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{d}{\frac{M \cdot \left(M \cdot h\right)}{d}}}\\ \mathbf{elif}\;M \leq 3.542576998845787 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h \cdot \left(M \cdot M\right)}} \cdot \frac{D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \log h\right)}}{{d}^{2}}\\ \end{array} \]

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
 (if (<= M -6.153227891404315e+162)
   (* 0.25 (/ (pow D 2.0) (/ d (/ (* M (* M h)) d))))
   (if (<= M 3.542576998845787e+151)
     (* 0.25 (* (/ D (/ d (* h (* M M)))) (/ D d)))
     (* 0.25 (/ (exp (fma 2.0 (log (* M D)) (log h))) (pow d 2.0))))))

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 tmp;
	if (M <= -6.153227891404315e+162) {
		tmp = 0.25 * (pow(D, 2.0) / (d / ((M * (M * h)) / d)));
	} else if (M <= 3.542576998845787e+151) {
		tmp = 0.25 * ((D / (d / (h * (M * M)))) * (D / d));
	} else {
		tmp = 0.25 * (exp(fma(2.0, log(M * D), log(h))) / pow(d, 2.0));
	}
	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 3 regimes

2. if M < -6.15322789140431471e162

   1. Initial program 64.0

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

   2. Taylor expanded in c0 around -inf 64.0

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

   3. Taylor expanded in c0 around 0 64.0

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

   4. Applied associate-/l*_binary64 64.0

      \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

   5. Simplified 64.0

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

   6. Applied associate-*r*_binary64 51.8

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

   if -6.15322789140431471e162 < M < 3.5425769988457872e151

   1. Initial program 58.9

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

   2. Taylor expanded in c0 around -inf 38.8

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

   3. Taylor expanded in c0 around 0 31.0

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

   4. Applied associate-/l*_binary64 30.8

      \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

   5. Simplified 27.8

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

   6. Applied associate-/r/_binary64 27.8

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

   7. Applied unpow2_binary64 27.8

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

   8. Applied times-frac_binary64 21.9

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

   if 3.5425769988457872e151 < M

   1. Initial program 64.0

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

   2. Taylor expanded in c0 around -inf 63.5

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

   3. Taylor expanded in c0 around 0 63.5

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

   4. Applied add-exp-log_binary64 63.8

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \color{blue}{e^{\log h}}\right)}{{d}^{2}} \]

   5. Applied pow-to-exp_binary64 63.8

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left(\color{blue}{e^{\log M \cdot 2}} \cdot e^{\log h}\right)}{{d}^{2}} \]

   6. Applied prod-exp_binary64 57.7

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{e^{\log M \cdot 2 + \log h}}}{{d}^{2}} \]

   7. Applied pow-to-exp_binary64 60.6

      \[\leadsto 0.25 \cdot \frac{\color{blue}{e^{\log D \cdot 2}} \cdot e^{\log M \cdot 2 + \log h}}{{d}^{2}} \]

   8. Applied prod-exp_binary64 54.8

      \[\leadsto 0.25 \cdot \frac{\color{blue}{e^{\log D \cdot 2 + \left(\log M \cdot 2 + \log h\right)}}}{{d}^{2}} \]

   9. Simplified 54.7

      \[\leadsto 0.25 \cdot \frac{e^{\color{blue}{\mathsf{fma}\left(2, \log \left(D \cdot M\right), \log h\right)}}}{{d}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.153227891404315 \cdot 10^{+162}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2}}{\frac{d}{\frac{M \cdot \left(M \cdot h\right)}{d}}}\\ \mathbf{elif}\;M \leq 3.542576998845787 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h \cdot \left(M \cdot M\right)}} \cdot \frac{D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \log h\right)}}{{d}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(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))))))