Average Error: 13.3 → 7.8
Time: 2.4m
Precision: 64
Internal Precision: 384
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} = -\infty:\\ \;\;\;\;\sqrt{1 - \left(M \cdot \left(\left(h \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \cdot w0\\ \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \le 1.2984016655051126 \cdot 10^{+304}:\\ \;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(M \cdot \left(\left(h \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \cdot w0\\ \end{array}\]

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 2 regimes

  2. if (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))) < -inf.0 or 1.2984016655051126e+304 < (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)))))

    1. Initial program 60.8

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

    3. Applied div-inv60.8

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

    4. Applied associate-*r*43.5

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

    6. Applied unpow243.5

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

    7. Applied associate-*l*36.0

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

    8. Taylor expanded around 0 39.6

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

    9. Applied simplify34.1

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

    11. Applied div-inv34.1

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

    12. Applied associate-*l*35.1

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

    13. Applied simplify35.3

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

    if -inf.0 < (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))) < 1.2984016655051126e+304

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied sqrt-prod0.2

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

    5. Applied associate-*r*0.2

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))