Average Error: 26.4 → 14.1
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le -1.3379212855191511 \cdot 10^{-27}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le -1.2781696294214 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right)\\ \mathbf{elif}\;h \le 1.4032693043090215 \cdot 10^{-131}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

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 h < -1.3379212855191511e-27

    1. Initial program 25.2

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

    2. Initial simplification24.9

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

    4. Applied associate-*l/24.9

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

    5. Applied associate-/r/22.2

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

    7. Applied times-frac21.0

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

    8. Applied associate-*l*21.8

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

    9. Taylor expanded around -inf 18.3

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

    10. Simplified14.5

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

    if -1.3379212855191511e-27 < h < -1.2781696294214e-310

    1. Initial program 28.5

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

    2. Initial simplification29.2

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

    4. Applied associate-*l/29.2

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

    5. Applied associate-/r/30.5

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

    7. Applied times-frac28.7

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

    8. Applied associate-*l*25.8

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

    9. Taylor expanded around -inf 18.1

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

    10. Simplified14.0

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

    if -1.2781696294214e-310 < h < 1.4032693043090215e-131

    1. Initial program 32.1

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

    2. Initial simplification32.9

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

    4. Applied associate-*l/32.9

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

    5. Applied associate-/r/34.0

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

    7. Applied times-frac32.7

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

    8. Applied associate-*l*29.9

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

    9. Taylor expanded around inf 17.2

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

    10. Simplified12.6

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

    if 1.4032693043090215e-131 < h

    1. Initial program 23.9

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

    2. Initial simplification24.1

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

    4. Applied associate-*l/24.1

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

    5. Applied associate-/r/22.0

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

    7. Applied times-frac20.7

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

    8. Applied associate-*l*20.9

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

    10. Applied div-inv20.9

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

    11. Applied unpow-prod-down14.3

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -1.3379212855191511 \cdot 10^{-27}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;h \le -1.2781696294214 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right)\\ \mathbf{elif}\;h \le 1.4032693043090215 \cdot 10^{-131}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{{h}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell} \cdot \left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{2} \cdot h\right)\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018230 
(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)))))