Average Error: 25.7 → 18.2
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -3.627163193721708 \cdot 10^{-35}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)\right)\\ \mathbf{elif}\;d \le -2.1882254971755 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \sqrt{\frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \sqrt{\frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{\frac{1}{2}}\right)\right)\\ \end{array}\]

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 3 regimes

  2. if d < -3.627163193721708e-35

    1. Initial program 20.7

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

    3. Applied associate-*r/19.8

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

    4. Taylor expanded around -inf 15.5

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

    5. Simplified11.3

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

    if -3.627163193721708e-35 < d < -2.1882254971755e-310

    1. Initial program 33.1

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

    2. Taylor expanded around -inf 29.3

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

    3. Simplified26.7

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

    if -2.1882254971755e-310 < d

    1. Initial program 25.5

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

    3. Applied associate-*r/24.7

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

    5. Applied add-sqr-sqrt24.7

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

    7. Applied div-inv24.7

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

    8. Applied unpow-prod-down18.6

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

    9. Simplified18.6

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -3.627163193721708 \cdot 10^{-35}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)\right)\\ \mathbf{elif}\;d \le -2.1882254971755 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\sqrt{\frac{-1}{\ell}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \sqrt{\frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}} \cdot \sqrt{\frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{\frac{1}{2}}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes25.018.29.415.643.9%
herbie shell --seed 2018286 +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)))))