Average Error: 13.4 → 7.2
Time: 3.1m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot h\right)\right) \cdot \frac{1}{\ell}} \le -2.881556634241017 \cdot 10^{+298}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\frac{M}{2} \cdot \left(-\frac{D}{d}\right)\right) \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{1}{2} \cdot D\right)\right) \cdot \frac{h}{\ell}\right) + 1)_*}\\ \mathbf{if}\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot h\right)\right) \cdot \frac{1}{\ell}} \le 1.5490384640270792 \cdot 10^{+303}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\frac{-\frac{1}{2} \cdot M}{\frac{d}{D}}\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) + 1)_*}\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if (* w0 (sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (* (* M D) (/ 1 (* 2 d))) h)) (/ 1 l))))) < -2.881556634241017e+298

    1. Initial program 52.1

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

    3. Applied div-inv52.1

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

    4. Applied associate-*r*56.0

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

    6. Applied unpow256.0

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

    7. Applied associate-*l*55.9

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

    8. Taylor expanded around 0 55.9

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

    9. Applied simplify46.9

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

    if -2.881556634241017e+298 < (* w0 (sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (* (* M D) (/ 1 (* 2 d))) h)) (/ 1 l))))) < 1.5490384640270792e+303

    1. Initial program 6.4

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

    3. Applied div-inv6.4

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

    4. Applied associate-*r*2.0

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

    6. Applied unpow22.0

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

    7. Applied associate-*l*0.3

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

    9. Applied div-inv0.3

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

    if 1.5490384640270792e+303 < (* w0 (sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (* (* M D) (/ 1 (* 2 d))) h)) (/ 1 l)))))

    1. Initial program 53.6

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

    3. Applied div-inv53.6

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

    4. Applied associate-*r*57.9

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

    6. Applied unpow257.9

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

    7. Applied associate-*l*57.8

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

    8. Taylor expanded around 0 57.8

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

    9. Applied simplify45.7

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018198 +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))))))