Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 28.5

Time: 21.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \leq -1.360662088649393 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \mathsf{fma}\left(\log d, -2, \log h\right)\right)}\\ \mathbf{elif}\;M \leq 7.0868357241638995 \cdot 10^{+140}:\\ \;\;\;\;\left(0.25 \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot \frac{h \cdot {M}^{2}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\\ \end{array} \]

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 -1.360662088649393e+154)
   (* 0.25 (exp (fma 2.0 (log (* M D)) (fma (log d) -2.0 (log h)))))
   (if (<= M 7.0868357241638995e+140)
     (* (* 0.25 (* D (/ D d))) (/ (* h (pow M 2.0)) d))
     (* (* 0.25 (/ (pow D 2.0) d)) (/ (* M (* M h)) 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 <= -1.360662088649393e+154) {
		tmp = 0.25 * exp(fma(2.0, log(M * D), fma(log(d), -2.0, log(h))));
	} else if (M <= 7.0868357241638995e+140) {
		tmp = (0.25 * (D * (D / d))) * ((h * pow(M, 2.0)) / d);
	} else {
		tmp = (0.25 * (pow(D, 2.0) / d)) * ((M * (M * h)) / 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 < -1.3606620886493931e154

   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 64.0

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

   3. Taylor expanded in c0 around 0 64.0

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

   4. Applied pow-to-exp_binary64 64.0

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

   5. Applied add-exp-log_binary64 64.0

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

   6. Applied pow-to-exp_binary64 64.0

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

   7. Applied prod-exp_binary64 64.0

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{e^{\log M \cdot 2 + \log h}}}{e^{\log d \cdot 2}} \]

   8. Applied pow-to-exp_binary64 64.0

      \[\leadsto 0.25 \cdot \frac{\color{blue}{e^{\log D \cdot 2}} \cdot e^{\log M \cdot 2 + \log h}}{e^{\log d \cdot 2}} \]

   9. Applied prod-exp_binary64 64.0

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

   10. Applied div-exp_binary64 64.0

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

   11. Simplified 58.2

       \[\leadsto 0.25 \cdot e^{\color{blue}{\mathsf{fma}\left(2, \log \left(D \cdot M\right), \mathsf{fma}\left(\log d, -2, \log h\right)\right)}} \]

   if -1.3606620886493931e154 < M < 7.0868357241638995e140

   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 38.5

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

   3. Taylor expanded in c0 around 0 31.1

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

   4. Applied unpow2_binary64 31.1

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

   5. Applied times-frac_binary64 27.3

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

   6. Applied associate-*r*_binary64 27.3

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

   7. Applied *-un-lft-identity_binary64 27.3

      \[\leadsto \left(0.25 \cdot \frac{{D}^{2}}{\color{blue}{1 \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{d} \]

   8. Applied add-sqr-sqrt_binary64 46.4

      \[\leadsto \left(0.25 \cdot \frac{{\color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)}}^{2}}{1 \cdot d}\right) \cdot \frac{{M}^{2} \cdot h}{d} \]

   9. Applied unpow-prod-down_binary64 46.4

      \[\leadsto \left(0.25 \cdot \frac{\color{blue}{{\left(\sqrt{D}\right)}^{2} \cdot {\left(\sqrt{D}\right)}^{2}}}{1 \cdot d}\right) \cdot \frac{{M}^{2} \cdot h}{d} \]

   10. Applied times-frac_binary64 44.5

       \[\leadsto \left(0.25 \cdot \color{blue}{\left(\frac{{\left(\sqrt{D}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{D}\right)}^{2}}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{d} \]

   11. Simplified 44.5

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

   12. Simplified 23.9

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

   if 7.0868357241638995e140 < 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 62.6

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

   3. Taylor expanded in c0 around 0 61.6

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

   4. Applied unpow2_binary64 61.6

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

   5. Applied times-frac_binary64 61.3

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

   6. Applied associate-*r*_binary64 61.3

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

   7. Applied sqr-pow_binary64 61.3

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

   8. Applied associate-*l*_binary64 49.8

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

   9. Simplified 49.8

      \[\leadsto \left(0.25 \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{{M}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(h \cdot M\right)}}{d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.360662088649393 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), \mathsf{fma}\left(\log d, -2, \log h\right)\right)}\\ \mathbf{elif}\;M \leq 7.0868357241638995 \cdot 10^{+140}:\\ \;\;\;\;\left(0.25 \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot \frac{h \cdot {M}^{2}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2022076 
(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))))))