Average Error: 13.4 → 7.8
Time: 3.3m
Precision: 64
Internal Precision: 384
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\frac{M}{d} \cdot \left(\left(h \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \le 5.470659647461577 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{1 - \left(\frac{M}{d} \cdot \left(\left(h \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \cdot w0\\ \mathbf{if}\;\sqrt{1 - \left(\frac{M}{d} \cdot \left(\left(h \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \le +\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{1}{\ell}}}\\ \end{array}\]

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. Split input into 3 regimes

  2. if (sqrt (- 1 (* (* (/ M d) (* (* h D) 1/2)) (/ (* (/ D 2) (/ M d)) l)))) < 5.470659647461577e+76

    1. Initial program 6.0

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

    3. Applied div-inv6.0

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

    4. Applied associate-*r*2.1

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

    6. Applied unpow22.1

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

    7. Applied associate-*l*1.1

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

    8. Taylor expanded around 0 2.0

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

    9. Applied simplify0.3

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

    if 5.470659647461577e+76 < (sqrt (- 1 (* (* (/ M d) (* (* h D) 1/2)) (/ (* (/ D 2) (/ M d)) l)))) < +inf.0

    1. Initial program 47.8

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

    3. Applied unpow247.8

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

    4. Applied associate-*l*44.4

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

    if +inf.0 < (sqrt (- 1 (* (* (/ M d) (* (* h D) 1/2)) (/ (* (/ D 2) (/ M d)) l))))

    1. Initial program 18.7

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

    3. Applied div-inv18.7

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

    4. Applied associate-*r*9.9

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

    6. Applied unpow29.9

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

    7. Applied associate-*l*4.5

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

    9. Applied add-cube-cbrt4.5

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

    10. Applied associate-*r*4.5

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))