Henrywood and Agarwal, Equation (13)

Average Error: 59.8 → 29.2

Time: 17.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(M \cdot M\right) \cdot h\\ \mathbf{if}\;c0 \leq -1.208739675252761 \cdot 10^{-87}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{D \cdot D}{d}\\ \mathbf{if}\;c0 \leq 3.803592874752181 \cdot 10^{-271}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot t_1}{d}\\ \mathbf{elif}\;c0 \leq 1.5768310252267593 \cdot 10^{+165}:\\ \;\;\;\;0.25 \cdot \frac{\left(t_0 \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot t_1\right)}{d}\\ \end{array}\\ \end{array} \]

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
 (let* ((t_0 (* (* M M) h)))
   (if (<= c0 -1.208739675252761e-87)
     (* 0.25 (* t_0 (* (/ D d) (/ D d))))
     (let* ((t_1 (/ (* D D) d)))
       (if (<= c0 3.803592874752181e-271)
         (* 0.25 (/ (* (* M (* M h)) t_1) d))
         (if (<= c0 1.5768310252267593e+165)
           (*
            0.25
            (/ (* (* t_0 (/ D (* (cbrt d) (cbrt d)))) (/ D (cbrt d))) d))
           (* 0.25 (/ (* h (* (* M M) t_1)) 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 t_0 = (M * M) * h;
	double tmp;
	if (c0 <= -1.208739675252761e-87) {
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)));
	} else {
		double t_1 = (D * D) / d;
		double tmp_1;
		if (c0 <= 3.803592874752181e-271) {
			tmp_1 = 0.25 * (((M * (M * h)) * t_1) / d);
		} else if (c0 <= 1.5768310252267593e+165) {
			tmp_1 = 0.25 * (((t_0 * (D / (cbrt(d) * cbrt(d)))) * (D / cbrt(d))) / d);
		} else {
			tmp_1 = 0.25 * ((h * ((M * M) * t_1)) / d);
		}
		tmp = tmp_1;
	}
	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 c0 < -1.208739675252761e-87

   1. Initial program 60.2

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

   2. Taylor expanded in c0 around -inf 44.0

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

   3. Taylor expanded in c0 around 0 36.1

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

   4. Applied unpow2_binary64 36.1

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

   5. Applied associate-/r*_binary64 33.7

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

   6. Simplified 33.1

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

   7. Applied *-un-lft-identity_binary64 33.1

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

   8. Applied times-frac_binary64 33.9

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

   9. Simplified 33.9

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

   10. Simplified 28.9

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

   if -1.208739675252761e-87 < c0 < 3.80359287475218101e-271

   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 in c0 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. Taylor expanded in c0 around 0 33.3

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

   4. Applied unpow2_binary64 33.3

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

   5. Applied associate-/r*_binary64 31.8

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

   6. Simplified 31.5

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

   7. Applied associate-*r*_binary64 29.8

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

   if 3.80359287475218101e-271 < c0 < 1.57683102522675927e165

   1. Initial program 59.2

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

   2. Taylor expanded in c0 around -inf 39.7

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

   3. Taylor expanded in c0 around 0 35.6

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

   4. Applied unpow2_binary64 35.6

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

   5. Applied associate-/r*_binary64 32.9

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

   6. Simplified 32.6

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

   7. Applied add-cube-cbrt_binary64 32.6

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

   8. Applied times-frac_binary64 29.5

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

   9. Applied associate-*r*_binary64 28.0

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

   if 1.57683102522675927e165 < c0

   1. Initial program 61.5

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

   2. Taylor expanded in c0 around -inf 50.0

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

   3. Taylor expanded in c0 around 0 37.4

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

   4. Applied unpow2_binary64 37.4

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

   5. Applied associate-/r*_binary64 34.1

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

   6. Simplified 33.4

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

   7. Applied associate-*l*_binary64 32.1

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

4. Final simplification 29.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.208739675252761 \cdot 10^{-87}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{elif}\;c0 \leq 3.803592874752181 \cdot 10^{-271}:\\ \;\;\;\;0.25 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D \cdot D}{d}}{d}\\ \mathbf{elif}\;c0 \leq 1.5768310252267593 \cdot 10^{+165}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right)}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2021291 
(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))))))