Average Error: 13.8 → 7.6
Time: 2.7m
Precision: 64
Internal Precision: 384
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{(\left(\frac{M}{2} \cdot \left(-\frac{D}{d}\right)\right) \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{1}{2} \cdot D\right)\right) \cdot \frac{h}{\ell}\right) + 1)_*} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{if}\;w0 \cdot \sqrt{(\left(\frac{M}{2} \cdot \left(-\frac{D}{d}\right)\right) \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{1}{2} \cdot D\right)\right) \cdot \frac{h}{\ell}\right) + 1)_*} \le 3.9756377647635456 \cdot 10^{+260}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\frac{M}{2} \cdot \left(-\frac{D}{d}\right)\right) \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{1}{2} \cdot D\right)\right) \cdot \frac{h}{\ell}\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

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 (fma (* (/ M 2) (- (/ D d))) (* (* (/ M d) (* 1/2 D)) (/ h l)) 1))) < -inf.0 or 3.9756377647635456e+260 < (* w0 (sqrt (fma (* (/ M 2) (- (/ D d))) (* (* (/ M d) (* 1/2 D)) (/ h l)) 1)))

    1. Initial program 50.3

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

    3. Applied div-inv50.3

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

    4. Applied associate-*r*33.3

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

    6. Applied unpow233.3

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

    7. Applied associate-*l*30.4

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

    if -inf.0 < (* w0 (sqrt (fma (* (/ M 2) (- (/ D d))) (* (* (/ M d) (* 1/2 D)) (/ h l)) 1))) < 3.9756377647635456e+260

    1. Initial program 2.6

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

    3. Applied div-inv2.6

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

    4. Applied associate-*r*3.1

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

    6. Applied unpow23.1

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

    7. Applied associate-*l*2.1

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

    8. Taylor expanded around 0 2.1

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

    9. Applied simplify0.6

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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))))))