Average Error: 13.5 → 8.5
Time: 1.2m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{\frac{D \cdot M}{2 \cdot d}}{\frac{\ell}{\frac{D \cdot M}{2 \cdot d}}}}{\sqrt[3]{\frac{1}{h}}} \cdot \frac{1}{\sqrt[3]{\frac{1}{h}} \cdot \sqrt[3]{\frac{1}{h}}}} \cdot w0\]

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 13.5

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

  2. Initial simplification13.1

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

  4. Applied div-inv13.1

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

  5. Applied associate-/r*10.1

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

  7. Applied add-cube-cbrt10.1

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

  8. Applied *-un-lft-identity10.1

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

  9. Applied times-frac10.1

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

  11. Applied associate-/l*8.5

    \[\leadsto \sqrt{1 - \frac{1}{\sqrt[3]{\frac{1}{h}} \cdot \sqrt[3]{\frac{1}{h}}} \cdot \frac{\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}}}{\sqrt[3]{\frac{1}{h}}}} \cdot w0\]

  12. Final simplification8.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018216 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))