Henrywood and Agarwal, Equation (13)

Average Error: 59.3 → 27.0

Time: 32.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \leq -1.0014621651146908 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq -1.2162146492768503 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{h}{\sqrt[3]{d}}}{d}\\ \mathbf{elif}\;D \leq 5.81954129307857 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(M \cdot M\right) + \left(\left(\log D + \log D\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array}\]

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 (<= D -1.0014621651146908e+154)
   (* 0.25 (/ (* (* M M) (/ (* D (* D h)) d)) d))
   (if (<= D -1.2162146492768503e+85)
     (*
      0.25
      (/ (* (* (* M M) (/ (* D D) (* (cbrt d) (cbrt d)))) (/ h (cbrt d))) d))
     (if (<= D 5.81954129307857e+153)
       (* 0.25 (/ (* M (* M (/ (* h (* D D)) d))) d))
       (*
        0.25
        (exp
         (-
          (+ (log (* M M)) (+ (+ (log D) (log 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 (D <= -1.0014621651146908e+154) {
		tmp = 0.25 * (((M * M) * ((D * (D * h)) / d)) / d);
	} else if (D <= -1.2162146492768503e+85) {
		tmp = 0.25 * ((((M * M) * ((D * D) / (cbrt(d) * cbrt(d)))) * (h / cbrt(d))) / d);
	} else if (D <= 5.81954129307857e+153) {
		tmp = 0.25 * ((M * (M * ((h * (D * D)) / d))) / d);
	} else {
		tmp = 0.25 * exp((log(M * M) + ((log(D) + log(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 4 regimes

2. if D < -1.0014621651146908e154

   1. Initial program 62.7

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

   2. Taylor expanded around -inf 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 associate-/r*_binary64_1045 64.0

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

   8. Simplified 64.0

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

   10. Applied associate-*l*_binary64_1042 47.1

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

   if -1.0014621651146908e154 < D < -1.2162146492768503e85

   1. Initial program 57.0

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

   2. Taylor expanded around -inf 46.2

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

   3. Simplified 48.3

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

   4. Taylor expanded around 0 41.3

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

   5. Simplified 41.3

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

   7. Applied associate-/r*_binary64_1045 37.8

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

   8. Simplified 36.1

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

   10. Applied add-cube-cbrt_binary64_1136 36.3

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

   11. Applied times-frac_binary64_1107 29.4

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

   12. Applied associate-*r*_binary64_1041 30.1

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

   if -1.2162146492768503e85 < D < 5.81954129307856981e153

   1. Initial program 59.0

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

   2. Taylor expanded around -inf 38.3

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

   3. Simplified 38.9

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

   4. Taylor expanded around 0 31.8

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

   5. Simplified 31.8

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

   7. Applied associate-/r*_binary64_1045 28.9

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

   8. Simplified 28.3

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

   10. Applied associate-*l*_binary64_1042 24.1

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

   if 5.81954129307856981e153 < D

   1. Initial program 62.6

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

   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_1139 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_1139 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_1139 64.0

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

   10. Applied add-exp-log_binary64_1139 64.0

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

   11. Applied prod-exp_binary64_1150 64.0

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

   12. Applied prod-exp_binary64_1150 56.2

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

   13. Applied add-exp-log_binary64_1139 56.2

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

   14. Applied prod-exp_binary64_1150 52.6

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

   15. Applied div-exp_binary64_1152 52.0

       \[\leadsto 0.25 \cdot \color{blue}{e^{\left(\log \left(M \cdot M\right) + \left(\left(\log D + \log D\right) + \log h\right)\right) - \log \left(d \cdot d\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.0014621651146908 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d}}{d}\\ \mathbf{elif}\;D \leq -1.2162146492768503 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{h}{\sqrt[3]{d}}}{d}\\ \mathbf{elif}\;D \leq 5.81954129307857 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(M \cdot \frac{h \cdot \left(D \cdot D\right)}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot e^{\left(\log \left(M \cdot M\right) + \left(\left(\log D + \log D\right) + \log h\right)\right) - \log \left(d \cdot d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021069 
(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))))))