Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 28.7

Time: 18.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -1.078861062403717 \cdot 10^{+126}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log \left(\frac{h}{d}\right) + 2 \cdot \log \left(M \cdot D\right)}}{d}\\ \mathbf{elif}\;M \leq 3.255830381164449 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(D \cdot D\right) + \left(\left(\log M + \log M\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array} \]

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 -1.078861062403717e+126)
   (* 0.25 (/ (exp (+ (log (/ h d)) (* 2.0 (log (* M D))))) d))
   (if (<= M 3.255830381164449e+75)
     (* 0.25 (/ (* D (* D (/ (* h (* M M)) d))) d))
     (*
      0.25
      (exp
       (- (+ (log (* D D)) (+ (+ (log M) (log M)) (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 <= -1.078861062403717e+126) {
		tmp = 0.25 * (exp(log(h / d) + (2.0 * log(M * D))) / d);
	} else if (M <= 3.255830381164449e+75) {
		tmp = 0.25 * ((D * (D * ((h * (M * M)) / d))) / d);
	} else {
		tmp = 0.25 * exp((log(D * D) + ((log(M) + log(M)) + 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 < -1.0788610624037171e126

   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 61.1

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

   3. Simplified 60.9

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

   4. Taylor expanded around 0 59.6

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

   5. Simplified 59.6

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

   7. Applied associate-/r*_binary64 59.2

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

   8. Simplified 59.0

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

   10. Applied add-exp-log_binary64 61.2

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

   11. Applied add-exp-log_binary64 62.4

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

   12. Applied add-exp-log_binary64 64.0

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

   13. Applied add-exp-log_binary64 64.0

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

   14. Applied prod-exp_binary64 64.0

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

   15. Applied prod-exp_binary64 64.0

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

   16. Applied div-exp_binary64 64.0

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

   17. Applied add-exp-log_binary64 64.0

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

   18. Applied add-exp-log_binary64 64.0

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

   19. Applied prod-exp_binary64 64.0

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

   20. Applied prod-exp_binary64 64.0

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

   21. Simplified 52.5

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

   if -1.0788610624037171e126 < M < 3.2558303811644489e75

   1. Initial program 58.5

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

   2. Taylor expanded around -inf 37.9

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

   3. Simplified 40.1

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

   4. Taylor expanded around 0 30.8

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

   5. Simplified 30.8

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

   7. Applied associate-/r*_binary64 27.5

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

   8. Simplified 27.1

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

   10. Applied associate-*l*_binary64 23.1

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

   if 3.2558303811644489e75 < M

   1. Initial program 63.7

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

   2. Taylor expanded around -inf 56.9

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

   3. Simplified 56.4

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

   4. Taylor expanded around 0 53.7

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

   5. Simplified 53.7

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

   7. Applied add-exp-log_binary64 53.9

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

   8. Applied add-exp-log_binary64 58.5

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

   9. Applied add-exp-log_binary64 58.5

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

   10. Applied add-exp-log_binary64 58.5

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

   11. Applied prod-exp_binary64 58.5

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

   12. Applied prod-exp_binary64 55.9

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

   13. Applied add-exp-log_binary64 55.9

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

   14. Applied prod-exp_binary64 51.4

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

   15. Applied div-exp_binary64 50.3

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.078861062403717 \cdot 10^{+126}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log \left(\frac{h}{d}\right) + 2 \cdot \log \left(M \cdot D\right)}}{d}\\ \mathbf{elif}\;M \leq 3.255830381164449 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(D \cdot D\right) + \left(\left(\log M + \log M\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))))