Average Error: 59.6 → 33.8
Time: 12.8s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c0 \leq -5.548180656061797 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -1.3573790657687668 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;c0 \leq -5.548180656061797 \cdot 10^{-91}:\\
\;\;\;\;0\\

\mathbf{elif}\;c0 \leq -1.3573790657687668 \cdot 10^{-114}:\\
\;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot 2\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 (<= c0 -5.548180656061797e-91)
   0.0
   (if (<= c0 -1.3573790657687668e-114)
     (* (/ c0 (* 2.0 w)) (* (* (/ c0 (* w h)) (* (/ d D) (/ d D))) 2.0))
     0.0)))
double code(double c0, double w, double h, double D, double d, double M) {
	return ((double) ((c0 / ((double) (2.0 * w))) * ((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) + ((double) sqrt(((double) (((double) ((((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))) * (((double) (c0 * ((double) (d * d)))) / ((double) (((double) (w * h)) * ((double) (D * D))))))) - ((double) (M * M))))))))));
}
↓
double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -5.548180656061797e-91)) {
		tmp = 0.0;
	} else {
		double tmp_1;
		if ((c0 <= -1.3573790657687668e-114)) {
			tmp_1 = ((double) ((c0 / ((double) (2.0 * w))) * ((double) (((double) ((c0 / ((double) (w * h))) * ((double) ((d / D) * (d / D))))) * 2.0))));
		} else {
			tmp_1 = 0.0;
		}
		tmp = tmp_1;
	}
	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 c0 < -5.54818065606179662e-91 or -1.3573790657687668e-114 < c0

    1. Initial program Error: 59.6 bits

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

    2. Taylor expanded around inf Error: 35.4 bits

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]
    3. Using strategy rm

    4. Applied mul0-rgtError: 33.5 bits

      \[\leadsto \color{blue}{0}\]

    if -5.54818065606179662e-91 < c0 < -1.3573790657687668e-114

    1. Initial program Error: 58.8 bits

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

    3. Applied flip-+Error: 61.2 bits

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

    4. SimplifiedError: 43.9 bits

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

    5. SimplifiedError: 50.0 bits

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{M \cdot M}{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{\frac{c0}{w \cdot h} \cdot \frac{{d}^{4} \cdot c0}{{D}^{4} \cdot \left(w \cdot h\right)} - M \cdot M}}}\]

    6. Taylor expanded around 0 Error: 59.0 bits

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

    7. SimplifiedError: 50.9 bits

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot 2\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 33.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.548180656061797 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -1.3573790657687668 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))))