Average Error: 13.3 → 7.2
Time: 2.9m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}} \le 3.410142006508361 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\frac{M}{2} \cdot \left(-\frac{D}{d}\right)\right) \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{1}{2} \cdot D\right)\right) \cdot \frac{h}{\ell}\right) + 1)_*}\\ \end{array}\]

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 (sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l)))) < 3.410142006508361e+143

    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*1.9

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

    6. Applied unpow21.9

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

    7. Applied associate-*l*0.2

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

    if 3.410142006508361e+143 < (sqrt (- 1 (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (/ 1 l))))

    1. Initial program 56.3

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

    3. Applied div-inv56.3

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

    4. Applied associate-*r*59.3

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

    6. Applied unpow259.3

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

    7. Applied associate-*l*58.9

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

    8. Taylor expanded around 0 58.9

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

    9. Applied simplify48.7

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018208 +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))))))