Average Error: 25.8 → 20.9
Time: 4.8m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 5.744555384444346 \cdot 10^{-258}:\\ \;\;\;\;\left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 3.161821923353777 \cdot 10^{+174}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\left({\left(\left|\sqrt[3]{\frac{d}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{d}{\ell}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 8.188519636352656 \cdot 10^{+285}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (* (* (pow (/ d h) (/ 1 2)) (exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))) (- 1 (/ (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h) (* 2 l)))) < 5.744555384444346e-258

    1. Initial program 33.9

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

    2. Taylor expanded around -inf 32.3

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

    if 5.744555384444346e-258 < (* (* (pow (/ d h) (/ 1 2)) (exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))) (- 1 (/ (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h) (* 2 l)))) < 3.161821923353777e+174

    1. Initial program 8.6

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

    3. Applied associate-*l/8.6

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

    4. Applied frac-times6.2

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

    6. Applied add-sqr-sqrt6.2

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

    7. Applied unpow-prod-down6.4

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

    9. Applied add-cube-cbrt6.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\sqrt{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\color{blue}{\left(\sqrt[3]{\frac{d}{\ell}} \cdot \sqrt[3]{\frac{d}{\ell}}\right) \cdot \sqrt[3]{\frac{d}{\ell}}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)\]

    10. Applied sqrt-prod6.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\sqrt{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\sqrt{\sqrt[3]{\frac{d}{\ell}} \cdot \sqrt[3]{\frac{d}{\ell}}} \cdot \sqrt{\sqrt[3]{\frac{d}{\ell}}}\right)}}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)\]

    11. Applied simplify6.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\sqrt{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\color{blue}{\left|\sqrt[3]{\frac{d}{\ell}}\right|} \cdot \sqrt{\sqrt[3]{\frac{d}{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)\]

    if 3.161821923353777e+174 < (* (* (pow (/ d h) (/ 1 2)) (exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))) (- 1 (/ (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h) (* 2 l)))) < 8.188519636352656e+285

    1. Initial program 26.7

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

    3. Applied associate-*l/26.7

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

    4. Applied frac-times26.7

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

    5. Taylor expanded around -inf 7.2

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

    if 8.188519636352656e+285 < (* (* (pow (/ d h) (/ 1 2)) (exp (* 1/2 (- (log (/ -1 l)) (log (/ -1 d)))))) (- 1 (/ (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h) (* 2 l))))

    1. Initial program 30.2

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

    3. Applied associate-*l/30.2

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

    4. Applied frac-times29.6

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

    6. Applied div-inv29.6

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

    7. Applied unpow-prod-down24.6

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify20.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 5.744555384444346 \cdot 10^{-258}:\\ \;\;\;\;\left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 3.161821923353777 \cdot 10^{+174}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(\left({\left(\left|\sqrt[3]{\frac{d}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{d}{\ell}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le 8.188519636352656 \cdot 10^{+285}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{\ell \cdot 2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 4.8m)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))