Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 23.7

Time: 24.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot \left(M \cdot h\right)}{d}\\ t_1 := \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\\ \mathbf{if}\;M \cdot M \leq 2.5 \cdot 10^{-323}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot t_0\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 7.722402878706791 \cdot 10^{+305}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \frac{D \cdot D}{d}\right)\\ \end{array} \]

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
 (let* ((t_0 (/ (* M (* M h)) d)) (t_1 (/ D (* (cbrt d) (cbrt d)))))
   (if (<= (* M M) 2.5e-323)
     (* 0.25 (* t_1 (* (/ D (cbrt d)) t_0)))
     (if (<= (* M M) 7.722402878706791e+305)
       (* 0.25 (* t_1 (* (* h (/ D d)) (/ (* M M) (cbrt d)))))
       (* 0.25 (* t_0 (/ (* D 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 t_0 = (M * (M * h)) / d;
	double t_1 = D / (cbrt(d) * cbrt(d));
	double tmp;
	if ((M * M) <= 2.5e-323) {
		tmp = 0.25 * (t_1 * ((D / cbrt(d)) * t_0));
	} else if ((M * M) <= 7.722402878706791e+305) {
		tmp = 0.25 * (t_1 * ((h * (D / d)) * ((M * M) / cbrt(d))));
	} else {
		tmp = 0.25 * (t_0 * ((D * 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 (*.f64 M M) < 2.47033e-323

   1. Initial program 55.9

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

   2. Taylor expanded in c0 around -inf 31.1

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

   3. Applied add-sqr-sqrt_binary64 47.1

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

   4. Applied unpow-prod-down_binary64 47.1

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

   5. Applied times-frac_binary64 45.5

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

   6. Simplified 45.5

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

   7. Simplified 27.3

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

   8. Applied add-cube-cbrt_binary64 27.3

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

   9. Applied times-frac_binary64 24.3

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

   10. Applied associate-*l*_binary64 23.1

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

   11. Applied associate-*r*_binary64 21.8

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

   if 2.47033e-323 < (*.f64 M M) < 7.72240287870679086e305

   1. Initial program 61.1

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

   2. Taylor expanded in c0 around -inf 31.9

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

   3. Applied add-sqr-sqrt_binary64 48.4

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

   4. Applied unpow-prod-down_binary64 48.4

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

   5. Applied times-frac_binary64 46.6

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

   6. Simplified 46.6

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

   7. Simplified 28.4

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

   8. Applied add-cube-cbrt_binary64 28.5

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

   9. Applied times-frac_binary64 24.4

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

   10. Applied associate-*l*_binary64 22.4

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

   11. Applied add-cube-cbrt_binary64 22.5

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

   12. Applied times-frac_binary64 20.5

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

   13. Applied associate-*r*_binary64 18.6

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

   14. Simplified 17.8

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

   if 7.72240287870679086e305 < (*.f64 M 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 63.8

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

   3. Applied add-sqr-sqrt_binary64 63.9

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

   4. Applied unpow-prod-down_binary64 63.9

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

   5. Applied times-frac_binary64 63.9

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

   6. Simplified 63.9

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

   7. Simplified 63.8

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

   8. Applied associate-*r*_binary64 49.9

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

4. Final simplification 23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2.5 \cdot 10^{-323}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 7.722402878706791 \cdot 10^{+305}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{d} \cdot \frac{D \cdot D}{d}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))))