Average Error: 59.8 → 27.4
Time: 19.9s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -1.3603938329136146 \cdot 10^{-61} \lor \neg \left(M \leq 4.700540983952727 \cdot 10^{+104}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \end{array} \]
\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 -1.3603938329136146 \cdot 10^{-61} \lor \neg \left(M \leq 4.700540983952727 \cdot 10^{+104}\right):\\
\;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\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{h \cdot \left(M \cdot M\right)}{d}\right)\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 -1.3603938329136146e-61) (not (<= M 4.700540983952727e+104)))
   (* 0.25 (* (/ (* D D) d) (/ (* M (* M h)) d)))
   (*
    0.25
    (* (/ D (* (cbrt d) (cbrt d))) (* (/ D (cbrt d)) (/ (* h (* M M)) 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 <= -1.3603938329136146e-61) || !(M <= 4.700540983952727e+104)) {
		tmp = 0.25 * (((D * D) / d) * ((M * (M * h)) / d));
	} else {
		tmp = 0.25 * ((D / (cbrt(d) * cbrt(d))) * ((D / cbrt(d)) * ((h * (M * 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if M < -1.3603938329136146e-61 or 4.70054098395272706e104 < M

    1. Initial program 63.0

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

    2. Taylor expanded in c0 around -inf 51.7

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

    3. Taylor expanded in c0 around 0 46.9

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

    4. Applied add-sqr-sqrt_binary6454.9

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

    5. Applied unpow-prod-down_binary6454.9

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

    6. Applied times-frac_binary6453.9

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

    7. Simplified53.9

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

    8. Simplified44.8

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

    9. Applied associate-*r*_binary6438.4

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

    if -1.3603938329136146e-61 < M < 4.70054098395272706e104

    1. Initial program 58.3

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

    2. Taylor expanded in c0 around -inf 37.3

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

    3. Taylor expanded in c0 around 0 30.0

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

    4. Applied add-sqr-sqrt_binary6447.1

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

    5. Applied unpow-prod-down_binary6447.1

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

    6. Applied times-frac_binary6445.4

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

    7. Simplified45.4

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

    8. Simplified26.6

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

    9. Applied add-cube-cbrt_binary6426.7

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

    10. Applied times-frac_binary6423.7

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

    11. Applied associate-*l*_binary6422.1

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

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.3603938329136146 \cdot 10^{-61} \lor \neg \left(M \leq 4.700540983952727 \cdot 10^{+104}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022072 
(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))))))