Average Error: 13.5 → 8.0
Time: 1.6m
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}{\ell}} \cdot w0\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

  1. Initial program 13.5

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

  2. Simplified12.0

    \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0}\]
  3. Using strategy rm

  4. Applied associate-*l/8.0

    \[\leadsto \sqrt{1 - \color{blue}{\frac{h \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
  5. Using strategy rm

  6. Applied div-inv8.0

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

  7. Final simplification8.0

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))