Average Error: 59.8 → 27.7
Time: 17.0s
Precision: binary64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
↓
\[0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d}}{d} \]
\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)
↓
0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d}}{d}

(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
 (* 0.25 (/ (/ (* D (* D (* M (* M h)))) d) 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) {
	return 0.25 * (((D * (D * (M * (M * h)))) / d) / d);
}

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. Initial program 59.8

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

  2. Taylor expanded in c0 around -inf 36.3

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

  3. Applied unpow2_binary6436.3

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

  4. Applied associate-/r*_binary6433.7

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

  5. Applied unpow2_binary6433.7

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

  6. Applied associate-*l*_binary6431.3

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

  7. Simplified31.3

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

  8. Applied unpow2_binary6431.3

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

  9. Applied associate-*l*_binary6427.7

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

  10. Final simplification27.7

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

Reproduce

herbie shell --seed 2021334 
(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))))))