Average Error: 25.2 → 14.6
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.24936350623451 \cdot 10^{-132}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right)\\ \mathbf{elif}\;d \le 1.8883723593521 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \le 1.3670940904069087 \cdot 10^{+77} \lor \neg \left(d \le 1.1036356717417465 \cdot 10^{+290}\right):\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

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 d < -2.24936350623451e-132

    1. Initial program 21.2

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

    2. Initial simplification20.9

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

    4. Applied associate-/l*19.5

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

    6. Applied associate-*l/19.5

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

    7. Applied associate-/l/17.8

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

    8. Taylor expanded around -inf 14.7

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

    9. Simplified10.5

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

    if -2.24936350623451e-132 < d < 1.8883723593521e-310

    1. Initial program 36.5

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

    2. Initial simplification37.2

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

    4. Applied associate-/l*35.8

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

    6. Applied associate-*l/35.8

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

    7. Applied associate-/l/35.4

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

    8. Taylor expanded around -inf 29.4

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

    9. Simplified26.9

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

    if 1.8883723593521e-310 < d < 1.3670940904069087e+77 or 1.1036356717417465e+290 < d

    1. Initial program 25.8

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

    2. Initial simplification26.3

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

    4. Applied associate-/l*24.8

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

    6. Applied associate-*l/24.8

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

    7. Applied associate-/l/23.2

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

    9. Applied div-inv23.2

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

    10. Applied unpow-prod-down17.1

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

    if 1.3670940904069087e+77 < d < 1.1036356717417465e+290

    1. Initial program 24.7

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

    2. Initial simplification23.8

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

    4. Applied associate-/l*23.0

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

    6. Applied associate-*l/23.0

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

    7. Applied associate-/l/21.9

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

    9. Applied div-inv21.9

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

    10. Applied unpow-prod-down10.2

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

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.24936350623451 \cdot 10^{-132}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\frac{-1}{h}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}}\right)\\ \mathbf{elif}\;d \le 1.8883723593521 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left(\frac{\sqrt{\frac{-1}{\ell}}}{{\left(\frac{-1}{d}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \le 1.3670940904069087 \cdot 10^{+77} \lor \neg \left(d \le 1.1036356717417465 \cdot 10^{+290}\right):\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{2 \cdot \ell}{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}}\right) \cdot \left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018220 +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)))))