Average Error: 59.5 → 25.5
Time: 25.6s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -6.052953150875497 \cdot 10^{+209}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), -\log \left(\frac{d}{h}\right)\right)}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \end{array} \]
\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 -6.052953150875497 \cdot 10^{+209}:\\
\;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), -\log \left(\frac{d}{h}\right)\right)}}{d}\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\


\end{array}

(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 -6.052953150875497e+209)
   (* 0.25 (/ (exp (fma 2.0 (log (* M D)) (- (log (/ d h))))) d))
   (*
    0.25
    (/ (* (* (* M (* M h)) (/ D (* (cbrt d) (cbrt d)))) (/ D (cbrt d))) d))))
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 <= -6.052953150875497e+209) {
		tmp = 0.25 * (exp(fma(2.0, log(M * D), -log(d / h))) / d);
	} else {
		tmp = 0.25 * ((((M * (M * h)) * (D / (cbrt(d) * cbrt(d)))) * (D / cbrt(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 2 regimes

  2. if M < -6.05295315087549724e209

    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 unpow2_binary6464.0

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

    4. Applied associate-/r*_binary6464.0

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

    5. Simplified64.0

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

    6. Applied add-exp-log_binary6464.0

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

    7. Applied add-exp-log_binary6464.0

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

    8. Applied add-exp-log_binary6464.0

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

    9. Applied prod-exp_binary6464.0

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

    10. Applied div-exp_binary6464.0

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

    11. Applied add-exp-log_binary6464.0

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

    12. Applied add-exp-log_binary6464.0

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

    13. Applied prod-exp_binary6464.0

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

    14. Applied add-exp-log_binary6464.0

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

    15. Applied prod-exp_binary6464.0

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

    16. Applied prod-exp_binary6464.0

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

    17. Simplified51.4

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

    if -6.05295315087549724e209 < M

    1. Initial program 59.3

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

    2. Taylor expanded in c0 around -inf 35.0

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

    3. Applied unpow2_binary6435.0

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

    4. Applied associate-/r*_binary6432.2

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

    5. Simplified31.6

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

    6. Applied add-cube-cbrt_binary6431.7

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

    7. Applied times-frac_binary6428.4

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

    8. Applied associate-*r*_binary6426.9

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

    9. Applied associate-*r*_binary6424.3

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

    10. Simplified24.3

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.052953150875497 \cdot 10^{+209}:\\ \;\;\;\;0.25 \cdot \frac{e^{\mathsf{fma}\left(2, \log \left(M \cdot D\right), -\log \left(\frac{d}{h}\right)\right)}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{D}{\sqrt[3]{d}}}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2022055 
(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))))))