Average Error: 58.2 → 47.1
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\]

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

Derivation

  1. Initial program 58.2

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

  2. Simplified52.2

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

  3. Taylor expanded around 0 59.3

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

  4. Simplified52.7

    \[\leadsto \frac{\color{blue}{\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}{2} \cdot \frac{c0}{w}\]
  5. Using strategy rm

  6. Applied pow152.7

    \[\leadsto \frac{\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}{2} \cdot \color{blue}{{\left(\frac{c0}{w}\right)}^{1}}\]

  7. Applied pow152.7

    \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}{2}\right)}^{1}} \cdot {\left(\frac{c0}{w}\right)}^{1}\]

  8. Applied pow-prod-down52.7

    \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}{2} \cdot \frac{c0}{w}\right)}^{1}}\]

  9. Simplified47.1

    \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\right)}}^{1}\]

  10. Final simplification47.1

    \[\leadsto \frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\]

Reproduce

herbie shell --seed 2019068 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 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))))))