Average Error: 25.7 → 18.9
Time: 4.6m
Precision: 64
Internal Precision: 384
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.9922219596467234 \cdot 10^{+165}:\\ \;\;\;\;\left(e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;\ell \le 3.281408225347533 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{if}\;\ell \le 1.389708416198949 \cdot 10^{-86}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

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 < -1.9922219596467234e+165

    1. Initial program 29.1

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

    2. Taylor expanded around -inf 25.1

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

    if -1.9922219596467234e+165 < l < 3.281408225347533e-309

    1. Initial program 23.4

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

    3. Applied div-inv23.4

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

    4. Applied associate-*r*21.6

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

    5. Taylor expanded around -inf 19.7

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

    if 3.281408225347533e-309 < l < 1.389708416198949e-86

    1. Initial program 30.9

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

    3. Applied div-inv30.9

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

    4. Applied associate-*r*23.0

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

    6. Applied div-inv23.0

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

    7. Applied unpow-prod-down11.2

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

    if 1.389708416198949e-86 < l

    1. Initial program 24.8

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

    3. Applied div-inv24.9

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

    4. Applied unpow-prod-down17.7

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

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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)))))