Average Error: 13.4 → 8.1
Time: 49.4s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d}}{\ell} \cdot \left(\frac{\sqrt[3]{\frac{D \cdot M}{2 \cdot d}}}{\frac{1}{h}} \cdot \left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right)\right)} \cdot w0\]

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

Derivation

  1. Initial program 13.4

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

  2. Simplified13.0

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

  4. Applied div-inv13.0

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

  5. Applied times-frac8.0

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

  7. Applied div-inv8.0

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

  8. Applied add-cube-cbrt8.1

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

  9. Applied times-frac8.1

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

  10. Simplified8.1

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

  11. Final simplification8.1

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

Reproduce

herbie shell --seed 2019093 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))