Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 27.3

Time: 16.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -1.5961129530600776 \cdot 10^{+143}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}{d}\\ \mathbf{elif}\;M \leq 1.338118781860413 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\left(\log M + \log M\right) + \left(\log \left(D \cdot D\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array}\]

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
 (if (<= M -1.5961129530600776e+143)
   (* 0.25 (/ (* M (* M (/ (* (* D D) h) d))) d))
   (if (<= M 1.338118781860413e+154)
     (* 0.25 (/ (* (* M M) (/ (* D (* D h)) d)) d))
     (*
      0.25
      (exp
       (- (+ (+ (log M) (log M)) (+ (log (* D D)) (log h))) (log (* d d))))))))

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 <= -1.5961129530600776e+143) {
		tmp = 0.25 * ((M * (M * (((D * D) * h) / d))) / d);
	} else if (M <= 1.338118781860413e+154) {
		tmp = 0.25 * (((M * M) * ((D * (D * h)) / d)) / d);
	} else {
		tmp = 0.25 * exp(((log(M) + log(M)) + (log(D * D) + log(h))) - log(d * d));
	}
	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 < -1.5961129530600776e143

   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 around -inf 62.2

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

   3. Simplified 62.3

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

   4. Taylor expanded around 0 61.6

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

   5. Simplified 61.6

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot d}}\]
   6. Using strategy rm

   7. Applied associate-/r*_binary64 61.2

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

   8. Simplified 61.4

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{\left(D \cdot D\right) \cdot h}{d}}}{d}\]
   9. Using strategy rm

   10. Applied associate-*l*_binary64 39.5

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

   if -1.5961129530600776e143 < M < 1.3381187818604131e154

   1. Initial program 58.8

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

   2. Taylor expanded around -inf 37.5

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

   3. Simplified 38.3

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

   4. Taylor expanded around 0 30.8

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

   5. Simplified 30.8

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot d}}\]
   6. Using strategy rm

   7. Applied associate-/r*_binary64 27.7

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

   8. Simplified 27.1

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot \frac{\left(D \cdot D\right) \cdot h}{d}}}{d}\]
   9. Using strategy rm

   10. Applied associate-*l*_binary64 24.3

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

   if 1.3381187818604131e154 < 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 around -inf 64.0

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

   3. Simplified 64.0

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

   4. Taylor expanded around 0 64.0

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

   5. Simplified 64.0

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot d}}\]
   6. Using strategy rm

   7. Applied add-exp-log_binary64 64.0

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

   8. Applied add-exp-log_binary64 64.0

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

   9. Applied add-exp-log_binary64 64.0

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

   10. Applied prod-exp_binary64 64.0

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

   11. Applied add-exp-log_binary64 64.0

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

   12. Applied add-exp-log_binary64 64.0

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

   13. Applied prod-exp_binary64 64.0

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

   14. Applied prod-exp_binary64 54.2

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

   15. Applied div-exp_binary64 53.1

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

4. Final simplification 27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.5961129530600776 \cdot 10^{+143}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}{d}\\ \mathbf{elif}\;M \leq 1.338118781860413 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\left(\log M + \log M\right) + \left(\log \left(D \cdot D\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021148 
(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))))))