Average Error: 13.9 → 9.5
Time: 3.4m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{2} \cdot \frac{D \cdot h}{d \cdot \ell}\right) \le -4.1331890661198384 \cdot 10^{+296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{2} \cdot \frac{D \cdot h}{d \cdot \ell}\right) \le 5.5778630145147836 \cdot 10^{-98}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{2} \cdot \frac{D \cdot h}{d \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \log_* (1 + (e^{\sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{\frac{M}{\ell}}{\frac{2}{D}} \cdot \frac{h}{d}\right)}} - 1)^*)\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (* M D) (* 2 d)) (* (/ M 2) (/ (* D h) (* d l)))) < -4.1331890661198384e+296

    1. Initial program 51.3

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

    3. Applied div-inv51.3

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

    4. Applied associate-*r*48.5

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

    6. Applied unpow248.5

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

    7. Applied associate-*l*47.2

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

    if -4.1331890661198384e+296 < (* (/ (* M D) (* 2 d)) (* (/ M 2) (/ (* D h) (* d l)))) < 5.5778630145147836e-98

    1. Initial program 5.9

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

    3. Applied div-inv5.9

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

    4. Applied associate-*r*3.0

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

    6. Applied unpow23.0

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

    7. Applied associate-*l*1.4

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

    9. Applied associate-*l*0.3

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

    10. Applied simplify5.4

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

    12. Applied div-inv5.4

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

    13. Applied div-inv5.4

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

    14. Applied times-frac5.0

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

    15. Applied associate-*l*4.5

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

    16. Applied simplify0.9

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

    if 5.5778630145147836e-98 < (* (/ (* M D) (* 2 d)) (* (/ M 2) (/ (* D h) (* d l))))

    1. Initial program 21.4

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

    3. Applied div-inv21.4

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

    4. Applied associate-*r*13.7

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

    6. Applied unpow213.7

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

    7. Applied associate-*l*12.3

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

    9. Applied associate-*l*12.2

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

    10. Applied simplify18.6

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

    12. Applied log1p-expm1-u20.3

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))