Average Error: 13.7 → 8.3
Time: 56.3s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left|\sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\frac{\ell}{\frac{D \cdot M}{2 \cdot d}}}}\right| \cdot w0\]

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 13.7

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

  2. Initial simplification13.3

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

  4. Applied associate-/r/10.4

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

  6. Applied associate-/l*8.9

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

  8. Applied add-cube-cbrt9.0

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

  10. Applied add-sqr-sqrt9.0

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

  11. Applied rem-sqrt-square9.0

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

  12. Simplified8.3

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

  13. Final simplification8.3

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

Runtime

Time bar (total: 56.3s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))