Average Error: 48.7 → 34.9
Time: 16.2s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{c0 \cdot d}{D \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M \leq -1.1426100242627277 \cdot 10^{-140}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{d}{D} \cdot t_0\right)\right)\\ \mathbf{elif}\;M \leq -9.618862001490325 \cdot 10^{-152}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;M \leq -1.917101077054597 \cdot 10^{-308} \lor \neg \left(M \leq 5.887206475242185 \cdot 10^{-287}\right):\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{d}{\frac{D}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{-{M}^{2}}\\ \end{array} \]
\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}
t_0 := \frac{c0 \cdot d}{D \cdot \left(w \cdot h\right)}\\
t_1 := \frac{c0}{2 \cdot w}\\
\mathbf{if}\;M \leq -1.1426100242627277 \cdot 10^{-140}:\\
\;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{d}{D} \cdot t_0\right)\right)\\

\mathbf{elif}\;M \leq -9.618862001490325 \cdot 10^{-152}:\\
\;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{c0 \cdot {d}^{2}}\right)\\

\mathbf{elif}\;M \leq -1.917101077054597 \cdot 10^{-308} \lor \neg \left(M \leq 5.887206475242185 \cdot 10^{-287}\right):\\
\;\;\;\;t_1 \cdot \left(2 \cdot \frac{d}{\frac{D}{t_0}}\right)\\

\mathbf{else}:\\
\;\;\;\;t_1 \cdot \sqrt{-{M}^{2}}\\


\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
 (let* ((t_0 (/ (* c0 d) (* D (* w h)))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= M -1.1426100242627277e-140)
     (* t_1 (* 2.0 (* (/ d D) t_0)))
     (if (<= M -9.618862001490325e-152)
       (*
        t_1
        (* 0.5 (/ (* (pow D 2.0) (* w (* h (pow M 2.0)))) (* c0 (pow d 2.0)))))
       (if (or (<= M -1.917101077054597e-308)
               (not (<= M 5.887206475242185e-287)))
         (* t_1 (* 2.0 (/ d (/ D t_0))))
         (* t_1 (sqrt (- (pow M 2.0)))))))))
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 t_0 = (c0 * d) / (D * (w * h));
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (M <= -1.1426100242627277e-140) {
		tmp = t_1 * (2.0 * ((d / D) * t_0));
	} else if (M <= -9.618862001490325e-152) {
		tmp = t_1 * (0.5 * ((pow(D, 2.0) * (w * (h * pow(M, 2.0)))) / (c0 * pow(d, 2.0))));
	} else if ((M <= -1.917101077054597e-308) || !(M <= 5.887206475242185e-287)) {
		tmp = t_1 * (2.0 * (d / (D / t_0)));
	} else {
		tmp = t_1 * sqrt(-pow(M, 2.0));
	}
	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 4 regimes

  2. if M < -1.14261002426272765e-140

    1. Initial program 50.7

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

    2. Taylor expanded in c0 around inf 43.0

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

    3. Applied unpow2_binary6443.0

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

    4. Applied associate-*l*_binary6440.9

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

    5. Applied unpow2_binary6440.9

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

    6. Applied associate-*l*_binary6438.3

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

    7. Applied times-frac_binary6435.4

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

    if -1.14261002426272765e-140 < M < -9.61886200149032494e-152

    1. Initial program 46.5

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

    2. Taylor expanded in c0 around -inf 39.0

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

    if -9.61886200149032494e-152 < M < -1.9171010770545972e-308 or 5.8872064752421847e-287 < M

    1. Initial program 47.9

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

    2. Taylor expanded in c0 around inf 43.9

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

    3. Applied unpow2_binary6443.9

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

    4. Applied associate-*l*_binary6441.5

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

    5. Applied unpow2_binary6441.5

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

    6. Applied associate-*l*_binary6439.0

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

    7. Applied associate-/l*_binary6436.5

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

    8. Simplified34.6

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

    if -1.9171010770545972e-308 < M < 5.8872064752421847e-287

    1. Initial program 41.8

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

    2. Taylor expanded in c0 around 0 34.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\sqrt{-{M}^{2}}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification34.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.1426100242627277 \cdot 10^{-140}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{D} \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot h\right)}\right)\right)\\ \mathbf{elif}\;M \leq -9.618862001490325 \cdot 10^{-152}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;M \leq -1.917101077054597 \cdot 10^{-308} \lor \neg \left(M \leq 5.887206475242185 \cdot 10^{-287}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d}{\frac{D}{\frac{c0 \cdot d}{D \cdot \left(w \cdot h\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt{-{M}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))))