Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 27.0

Time: 21.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -1.7477617212624623 \cdot 10^{+148}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), -\log \left(\frac{d}{h}\right)\right)}}{d}\\ \mathbf{elif}\;M \leq 2.795396578891542 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left({D}^{2}\right) + \left(2 \cdot \log M + \log h\right)\right) - \log \left({d}^{2}\right)}\\ \end{array} \]

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 -1.7477617212624623e+148)
   (* 0.25 (/ (exp (fma 2.0 (log (* M D)) (- (log (/ d h))))) d))
   (if (<= M 2.795396578891542e+127)
     (*
      0.25
      (/ (* (* (* h (* M M)) (/ D (* (cbrt d) (cbrt d)))) (/ D (cbrt d))) d))
     (*
      0.25
      (exp
       (-
        (+ (log (pow D 2.0)) (+ (* 2.0 (log M)) (log h)))
        (log (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 <= -1.7477617212624623e+148) {
		tmp = 0.25 * (exp(fma(2.0, log(M * D), -log(d / h))) / d);
	} else if (M <= 2.795396578891542e+127) {
		tmp = 0.25 * ((((h * (M * M)) * (D / (cbrt(d) * cbrt(d)))) * (D / cbrt(d))) / d);
	} else {
		tmp = 0.25 * exp((log(pow(D, 2.0)) + ((2.0 * log(M)) + log(h))) - log(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 < -1.7477617212624623e148

   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 63.2

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

   3. Taylor expanded in c0 around 0 63.1

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

   4. Applied unpow2_binary64 63.1

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

   5. Applied associate-/r*_binary64 63.1

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

   6. Simplified 63.0

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

   7. Applied add-exp-log_binary64 63.6

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

   8. Applied add-exp-log_binary64 63.8

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

   9. Applied add-exp-log_binary64 63.8

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

   10. Applied prod-exp_binary64 63.8

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

   11. Applied div-exp_binary64 63.6

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

   12. Applied add-exp-log_binary64 64.0

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

   13. Applied add-exp-log_binary64 64.0

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

   14. Applied prod-exp_binary64 64.0

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

   15. Applied add-exp-log_binary64 64.0

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

   16. Applied prod-exp_binary64 64.0

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

   17. Applied prod-exp_binary64 64.0

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

   18. Simplified 52.9

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

   if -1.7477617212624623e148 < M < 2.7953965788915422e127

   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 38.7

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

   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 unpow2_binary64 31.0

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

   5. Applied associate-/r*_binary64 28.0

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

   6. Simplified 27.5

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

   7. Applied add-cube-cbrt_binary64 27.5

      \[\leadsto 0.25 \cdot \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}{d} \]

   8. Applied times-frac_binary64 24.1

      \[\leadsto 0.25 \cdot \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{D}{\sqrt[3]{d}}\right)}}{d} \]

   9. Applied associate-*r*_binary64 22.4

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}}{d} \]

   if 2.7953965788915422e127 < 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 61.3

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

   3. Taylor expanded in c0 around 0 59.7

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

   4. Applied add-exp-log_binary64 59.8

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

   5. Applied add-exp-log_binary64 61.5

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

   6. Applied pow-to-exp_binary64 61.5

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

   7. Applied prod-exp_binary64 57.7

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

   8. Applied add-exp-log_binary64 57.7

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

   9. Applied prod-exp_binary64 52.9

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

   10. Applied div-exp_binary64 51.8

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

4. Final simplification 27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.7477617212624623 \cdot 10^{+148}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), -\log \left(\frac{d}{h}\right)\right)}}{d}\\ \mathbf{elif}\;M \leq 2.795396578891542 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left({D}^{2}\right) + \left(2 \cdot \log M + \log h\right)\right) - \log \left({d}^{2}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))))