Average Error: 59.5 → 26.7
Time: 21.5s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq 6.497139624625351 \cdot 10^{-201} \lor \neg \left(M \leq 1.9619361743369715 \cdot 10^{+149}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{\left(D \cdot D\right) \cdot h}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \end{array}\]
\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 6.497139624625351 \cdot 10^{-201} \lor \neg \left(M \leq 1.9619361743369715 \cdot 10^{+149}\right):\\
\;\;\;\;0.25 \cdot \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{\left(D \cdot D\right) \cdot h}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)\right)\\

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

\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 (or (<= M 6.497139624625351e-201) (not (<= M 1.9619361743369715e+149)))
   (*
    0.25
    (* (/ M (* (cbrt d) (cbrt d))) (* (/ (* (* D D) h) d) (/ M (cbrt d)))))
   (* 0.25 (* (/ (* M M) d) (/ (* D (* D h)) 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 <= 6.497139624625351e-201) || !(M <= 1.9619361743369715e+149)) {
		tmp = 0.25 * ((M / (cbrt(d) * cbrt(d))) * ((((D * D) * h) / d) * (M / cbrt(d))));
	} else {
		tmp = 0.25 * (((M * M) / d) * ((D * (D * 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if M < 6.49713962462535073e-201 or 1.96193617433697147e149 < M

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 42.6

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

    3. Simplified43.3

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

    4. Taylor expanded around 0 36.9

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

    5. Simplified36.9

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot d}}\]
    6. Using strategy rm

    7. Applied times-frac_binary64_110733.9

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary64_113633.9

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

    10. Applied times-frac_binary64_110729.7

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

    11. Applied associate-*l*_binary64_104227.7

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

    12. Simplified27.7

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

    if 6.49713962462535073e-201 < M < 1.96193617433697147e149

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 38.0

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

    3. Simplified38.6

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

    4. Taylor expanded around 0 31.1

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

    5. Simplified31.1

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot h\right)}{d \cdot d}}\]
    6. Using strategy rm

    7. Applied times-frac_binary64_110728.0

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{\left(D \cdot D\right) \cdot h}{d}\right)}\]
    8. Using strategy rm

    9. Applied associate-*l*_binary64_104224.4

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.497139624625351 \cdot 10^{-201} \lor \neg \left(M \leq 1.9619361743369715 \cdot 10^{+149}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{\left(D \cdot D\right) \cdot h}{d} \cdot \frac{M}{\sqrt[3]{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))))))