Henrywood and Agarwal, Equation (13)

Average Error: 59.3 → 23.6

Time: 18.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ t_3 := t_0 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{if}\;t_2 \leq -7.130962801645547 \cdot 10^{-182}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(h \cdot M\right)\right) \cdot \frac{M}{d}\right)\\ \mathbf{elif}\;t_2 \leq 1.1298573201221567 \cdot 10^{+246}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \]

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 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))))
        (t_3 (* t_0 (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0)))))))
   (if (<= t_2 -7.130962801645547e-182)
     t_3
     (if (<= t_2 0.0)
       (* 0.25 (* (* (/ (* D D) d) (* h M)) (/ M d)))
       (if (<= t_2 1.1298573201221567e+246)
         t_3
         (*
          0.25
          (*
           (/ D (* (cbrt d) (cbrt d)))
           (* (/ D (cbrt d)) (/ (* M (* h M)) 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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt((t_1 * t_1) - (M * M)));
	double t_3 = t_0 * (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0))));
	double tmp;
	if (t_2 <= -7.130962801645547e-182) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = 0.25 * ((((D * D) / d) * (h * M)) * (M / d));
	} else if (t_2 <= 1.1298573201221567e+246) {
		tmp = t_3;
	} else {
		tmp = 0.25 * ((D / (cbrt(d) * cbrt(d))) * ((D / cbrt(d)) * ((M * (h * M)) / 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 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -7.13096280164554702e-182 or -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 1.12985732012215666e246

   1. Initial program 37.9

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

   2. Taylor expanded in c0 around inf 34.6

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

   if -7.13096280164554702e-182 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0

   1. Initial program 31.4

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

   2. Taylor expanded in c0 around -inf 26.8

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

   3. Applied add-sqr-sqrt_binary64 44.9

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

   4. Applied unpow-prod-down_binary64 44.9

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

   5. Applied times-frac_binary64 43.8

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

   6. Simplified 43.8

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

   7. Simplified 24.6

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

   8. Applied associate-*r*_binary64 22.4

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

   9. Applied *-un-lft-identity_binary64 22.4

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

   10. Applied times-frac_binary64 20.8

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

   11. Applied associate-*r*_binary64 18.6

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

   12. Simplified 18.6

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

   if 1.12985732012215666e246 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 63.9

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

   2. Taylor expanded in c0 around -inf 34.6

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

   3. Applied add-sqr-sqrt_binary64 49.3

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

   4. Applied unpow-prod-down_binary64 49.3

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

   5. Applied times-frac_binary64 47.7

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

   6. Simplified 47.7

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

   7. Simplified 31.1

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

   8. Applied associate-*r*_binary64 28.8

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

   9. Applied add-cube-cbrt_binary64 28.8

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

   10. Applied times-frac_binary64 25.0

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

   11. Applied associate-*l*_binary64 22.9

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

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -7.130962801645547 \cdot 10^{-182}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(h \cdot M\right)\right) \cdot \frac{M}{d}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 1.1298573201221567 \cdot 10^{+246}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))))))