Average Error: 13.3 → 8.6
Time: 52.7s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{\frac{M}{\frac{d}{D}}}{4} \cdot \left(h \cdot \frac{1}{\frac{\frac{d}{D}}{M}}\right)}{\ell}} \cdot w0\]

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

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.3

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

  2. Initial simplification13.0

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

  4. Applied associate-/r/10.3

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

  6. Applied associate-*l/10.3

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

  7. Simplified8.6

    \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{4}}}{\ell}} \cdot w0\]
  8. Using strategy rm

  9. Applied clear-num8.6

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

  10. Final simplification8.6

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

Runtime

Time bar (total: 52.7s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))