Average Error: 25.2 → 18.9
Time: 4.2m
Precision: 64
Internal Precision: 384
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.284360178812978 \cdot 10^{-69}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;\ell \le -1.102973268589884 \cdot 10^{-309}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{\frac{d}{h}} \cdot \sqrt[3]{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot e^{\left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right) \cdot \frac{1}{2}}\right) \cdot \left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{2 \cdot \ell}\right)\\ \mathbf{if}\;\ell \le 6.908580337186389 \cdot 10^{-69}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{2 \cdot \ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Split input into 4 regimes

  2. if l < -1.284360178812978e-69

    1. Initial program 24.3

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

    3. Applied times-frac24.6

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

    if -1.284360178812978e-69 < l < -1.102973268589884e-309

    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. Using strategy rm

    3. Applied associate-*l/28.5

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

    4. Applied frac-times22.7

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

    6. Applied add-cube-cbrt23.0

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

    7. Applied unpow-prod-down23.0

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

    8. Taylor expanded around -inf 15.0

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

    if -1.102973268589884e-309 < l < 6.908580337186389e-69

    1. Initial program 28.2

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

    3. Applied associate-*l/28.2

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

    4. Applied frac-times22.0

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

    6. Applied div-inv22.0

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

    7. Applied unpow-prod-down10.4

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

    if 6.908580337186389e-69 < l

    1. Initial program 24.2

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

    3. Applied div-inv24.2

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

    4. Applied unpow-prod-down17.2

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

  4. Applied simplify18.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\ell \le -1.284360178812978 \cdot 10^{-69}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;\ell \le -1.102973268589884 \cdot 10^{-309}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{\frac{d}{h}} \cdot \sqrt[3]{\frac{d}{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot e^{\left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right) \cdot \frac{1}{2}}\right) \cdot \left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{2 \cdot \ell}\right)\\ \mathbf{if}\;\ell \le 6.908580337186389 \cdot 10^{-69}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}{2 \cdot \ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(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)))))