Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 24.2

Time: 18.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\\ \mathbf{if}\;M \leq -3.660832784072471 \cdot 10^{+116}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)\right)\\ \mathbf{elif}\;M \leq 1.3559963451736178 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log \left({D}^{2}\right) + \left(2 \cdot \log M + \log h\right)}}{{d}^{2}}\\ \end{array} \]

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 (/ D (* (cbrt d) (cbrt d)))))
   (if (<= M -3.660832784072471e+116)
     (* 0.25 (* t_0 (* (/ D (cbrt d)) (/ (* M (* M h)) d))))
     (if (<= M 1.3559963451736178e+154)
       (* 0.25 (* t_0 (* (* h (/ D d)) (/ (* M M) (cbrt d)))))
       (*
        0.25
        (/
         (exp (+ (log (pow D 2.0)) (+ (* 2.0 (log M)) (log h))))
         (pow d 2.0)))))))

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 = D / (cbrt(d) * cbrt(d));
	double tmp;
	if (M <= -3.660832784072471e+116) {
		tmp = 0.25 * (t_0 * ((D / cbrt(d)) * ((M * (M * h)) / d)));
	} else if (M <= 1.3559963451736178e+154) {
		tmp = 0.25 * (t_0 * ((h * (D / d)) * ((M * M) / cbrt(d))));
	} else {
		tmp = 0.25 * (exp(log(pow(D, 2.0)) + ((2.0 * log(M)) + log(h))) / pow(d, 2.0));
	}
	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 < -3.66083278407247109e116

   1. Initial program 63.8

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

   2. Taylor expanded in c0 around -inf 58.0

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

   3. Applied add-sqr-sqrt_binary64 61.5

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

   4. Applied unpow-prod-down_binary64 61.5

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

   5. Applied times-frac_binary64 61.0

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

   6. Simplified 61.0

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

   7. Simplified 57.2

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

   8. Applied add-cube-cbrt_binary64 57.3

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

   9. Applied times-frac_binary64 56.6

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

   10. Applied associate-*l*_binary64 55.9

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

   11. Applied associate-*r*_binary64 41.7

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

   if -3.66083278407247109e116 < M < 1.35599634517361779e154

   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 30.8

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

   3. Applied add-sqr-sqrt_binary64 47.5

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

   4. Applied unpow-prod-down_binary64 47.5

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

   5. Applied times-frac_binary64 45.7

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

   6. Simplified 45.7

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

   7. Simplified 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 24.1

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

   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 21.7

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

   13. Applied associate-*r*_binary64 20.7

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

   14. Simplified 19.7

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

   if 1.35599634517361779e154 < 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 64.0

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

   3. Applied add-exp-log_binary64 64.0

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

   4. Applied pow-to-exp_binary64 64.0

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

   5. Applied prod-exp_binary64 58.3

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

   6. Applied add-exp-log_binary64 58.3

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

   7. Applied prod-exp_binary64 54.0

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.660832784072471 \cdot 10^{+116}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)\right)\\ \mathbf{elif}\;M \leq 1.3559963451736178 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{e^{\log \left({D}^{2}\right) + \left(2 \cdot \log M + \log h\right)}}{{d}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))))