Average Error: 59.6 → 27.1
Time: 26.2s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 5.123614575828501 \cdot 10^{+304}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{\sqrt{d}} \cdot \frac{M}{\frac{{d}^{1.5}}{M \cdot h}}\right)\\ \end{array} \]
\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 \cdot M \leq 5.123614575828501 \cdot 10^{+304}:\\
\;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)}{d}\\

\mathbf{else}:\\
\;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{\sqrt{d}} \cdot \frac{M}{\frac{{d}^{1.5}}{M \cdot h}}\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 (<= (* M M) 5.123614575828501e+304)
   (* 0.25 (/ (* D (* D (/ (* (* M M) h) d))) d))
   (* 0.25 (* (/ (* D D) (sqrt d)) (/ M (/ (pow d 1.5) (* M h)))))))
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 * M) <= 5.123614575828501e+304) {
		tmp = 0.25 * ((D * (D * (((M * M) * h) / d))) / d);
	} else {
		tmp = 0.25 * (((D * D) / sqrt(d)) * (M / (pow(d, 1.5) / (M * h))));
	}
	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 (*.f64 M M) < 5.12361457582850092e304

    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 around -inf 38.4

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

    3. Simplified40.2

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

    4. Taylor expanded around 0 30.7

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

    5. Simplified30.7

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

    7. Applied associate-/r*_binary6427.5

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

    8. Simplified26.9

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

    10. Applied associate-*l*_binary6422.8

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

    if 5.12361457582850092e304 < (*.f64 M 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 around -inf 63.9

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

    3. Simplified63.9

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

    4. Taylor expanded around 0 63.9

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

    5. Simplified63.9

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

    7. Applied associate-/r*_binary6463.9

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

    8. Simplified63.9

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

    10. Applied add-sqr-sqrt_binary6463.9

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

    11. Applied times-frac_binary6463.9

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

    12. Simplified52.7

      \[\leadsto 0.25 \cdot \left(\frac{D \cdot D}{\sqrt{d}} \cdot \color{blue}{\frac{M}{\frac{{d}^{1.5}}{M \cdot h}}}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 5.123614575828501 \cdot 10^{+304}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{\sqrt{d}} \cdot \frac{M}{\frac{{d}^{1.5}}{M \cdot h}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))))