Henrywood and Agarwal, Equation (13)

Average Error: 59.3 → 26.3

Time: 30.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -2.0235561551202397 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)}{\sqrt[3]{d}}\\ \mathbf{elif}\;M \leq 1.7096528760268854 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \frac{\frac{M \cdot M}{d} \cdot \left(D \cdot \left(D \cdot h\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{M}{d}\right)}{d}\\ \end{array}\]

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 -2.0235561551202397e+153)
   (*
    0.25
    (/ (* (* (* D D) h) (* (/ M d) (/ M (* (cbrt d) (cbrt d))))) (cbrt d)))
   (if (<= M 1.7096528760268854e+131)
     (* 0.25 (/ (* (/ (* M M) d) (* D (* D h))) d))
     (* 0.25 (/ (* M (* (* (* D D) h) (/ M 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 <= -2.0235561551202397e+153) {
		tmp = 0.25 * ((((D * D) * h) * ((M / d) * (M / (cbrt(d) * cbrt(d))))) / cbrt(d));
	} else if (M <= 1.7096528760268854e+131) {
		tmp = 0.25 * ((((M * M) / d) * (D * (D * h))) / d);
	} else {
		tmp = 0.25 * ((M * (((D * D) * h) * (M / 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 < -2.0235561551202397e153

   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 63.9

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

   3. Simplified 63.9

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

   4. Taylor expanded around 0 63.8

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

   5. Simplified 63.8

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

   7. Applied associate-/r*_binary64_1045 63.7

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

   8. Simplified 63.7

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

   10. Applied add-cube-cbrt_binary64_1136 63.7

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

   11. Applied associate-/r*_binary64_1045 63.7

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

   12. Simplified 35.9

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

   if -2.0235561551202397e153 < M < 1.70965287602688544e131

   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 37.6

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

   3. Simplified 38.4

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

   4. Taylor expanded around 0 30.9

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

   5. Simplified 30.9

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

   7. Applied associate-/r*_binary64_1045 27.9

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

   8. Simplified 27.3

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

   10. Applied associate-*l*_binary64_1042 24.7

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

   if 1.70965287602688544e131 < 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 61.1

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

   3. Simplified 61.0

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

   4. Taylor expanded around 0 59.9

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

   5. Simplified 59.9

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

   7. Applied associate-/r*_binary64_1045 59.7

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

   8. Simplified 58.6

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

   10. Applied *-un-lft-identity_binary64_1101 58.6

       \[\leadsto 0.25 \cdot \frac{\frac{M \cdot M}{\color{blue}{1 \cdot d}} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d}\]

   11. Applied times-frac_binary64_1107 41.5

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\frac{M}{1} \cdot \frac{M}{d}\right)} \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d}\]

   12. Applied associate-*l*_binary64_1042 35.8

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\frac{M}{1} \cdot \left(\frac{M}{d} \cdot \left(\left(D \cdot D\right) \cdot h\right)\right)}}{d}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.0235561551202397 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)}{\sqrt[3]{d}}\\ \mathbf{elif}\;M \leq 1.7096528760268854 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \frac{\frac{M \cdot M}{d} \cdot \left(D \cdot \left(D \cdot h\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot \frac{M}{d}\right)}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021097 
(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))))))