Average Error: 25.2 → 18.4
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 2.3436643645613 \cdot 10^{-310}:\\ \;\;\;\;\left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;\ell \le 4.634711685081233 \cdot 10^{+33}:\\ \;\;\;\;\left(1 - \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\ell \le 1.2612749137327032 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{-\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{\ell} \cdot \left(\frac{1}{4} \cdot h\right)}{\frac{d}{D}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) + {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot (\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot \left(\frac{h}{\ell}\right) + 1)_*}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array}\]

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 < 2.3436643645613e-310

    1. Initial program 25.0

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

    3. Applied div-inv25.0

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

    4. Applied associate-*r*24.4

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

    5. Taylor expanded around -inf 22.1

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

    6. Simplified18.6

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

    if 2.3436643645613e-310 < l < 4.634711685081233e+33

    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

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

    4. Applied associate-*r*21.3

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

    6. Applied div-inv21.3

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

    7. Applied unpow-prod-down13.5

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

    8. Simplified13.5

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

    if 4.634711685081233e+33 < l < 1.2612749137327032e+251

    1. Initial program 24.5

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

    3. Applied div-inv24.5

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

    4. Applied associate-*r*25.8

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

    6. Applied sub-neg25.8

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

    7. Applied distribute-lft-in25.7

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

    8. Simplified22.7

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

    10. Applied associate-*l/23.4

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

    11. Simplified23.0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 + \left(\left(-\sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\color{blue}{\frac{M}{\ell} \cdot \left(h \cdot \frac{1}{4}\right)}}{\frac{d}{D}}\right)\]

    if 1.2612749137327032e+251 < l

    1. Initial program 32.2

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

    2. Initial simplification31.8

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

    4. Applied sqrt-div24.6

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

    5. Applied sqrt-div17.2

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

    6. Applied frac-times17.1

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

    7. Applied associate-*r/16.8

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 2.3436643645613 \cdot 10^{-310}:\\ \;\;\;\;\left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;\ell \le 4.634711685081233 \cdot 10^{+33}:\\ \;\;\;\;\left(1 - \left(h \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\ell \le 1.2612749137327032 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{-\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{\ell} \cdot \left(\frac{1}{4} \cdot h\right)}{\frac{d}{D}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) + {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot (\left(\frac{-1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)\right) \cdot \left(\frac{h}{\ell}\right) + 1)_*}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes22.818.48.214.629.5%
herbie shell --seed 2018353 +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)))))