Average Error: 13.5 → 8.6

Time: 51.0s

Precision: 64

Internal Precision: 128

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{\frac{d}{D}}}{4} \cdot \left(\frac{M}{\frac{d}{D}} \cdot h\right)}{\ell}}\]

Error

The X axis uses an exponential scale — Bits error versus `w0`

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Derivation

Initial program 13.5
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
Initial simplification13.2
\[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
Using strategy rm
Applied associate-/r/10.0
\[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h}} \cdot w0\]
Using strategy rm
Applied associate-*l/10.2
\[\leadsto \sqrt{1 - \color{blue}{\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot h}{\ell}}} \cdot w0\]
Simplified8.6
\[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{4}}}{\ell}} \cdot w0\]
Final simplification8.6
\[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{\frac{d}{D}}}{4} \cdot \left(\frac{M}{\frac{d}{D}} \cdot h\right)}{\ell}}\]

Runtime

herbie shell --seed 2018277 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))