Average Error: 25.4 → 21.4
Time: 4.2m
Precision: 64
Internal Precision: 576
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \le 3.805530156414986 \cdot 10^{-17}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}}\right) \cdot \sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}}\right)\\ \mathbf{if}\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \le 1.9666897595934545 \cdot 10^{+282}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \end{array}\]

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 (* (* (* (pow d (/ 1 2)) (pow (/ 1 h) (/ 1 2))) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 3.805530156414986e-17

    1. Initial program 20.8

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

    3. Applied add-cube-cbrt21.0

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

    5. Applied div-inv21.0

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

    6. Applied unpow-prod-down15.9

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

    if 3.805530156414986e-17 < (* (* (* (pow d (/ 1 2)) (pow (/ 1 h) (/ 1 2))) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 1.9666897595934545e+282

    1. Initial program 12.5

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

    3. Applied div-inv12.5

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

    4. Applied unpow-prod-down0.5

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

    if 1.9666897595934545e+282 < (* (* (* (pow d (/ 1 2)) (pow (/ 1 h) (/ 1 2))) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

    1. Initial program 30.7

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

    3. Applied associate-*l/30.7

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

    4. Applied frac-times29.2

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

  4. Applied simplify21.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \le 3.805530156414986 \cdot 10^{-17}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}}\right) \cdot \sqrt[3]{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}}\right)\\ \mathbf{if}\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \le 1.9666897595934545 \cdot 10^{+282}:\\ \;\;\;\;\left(\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot h}{\ell \cdot 2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018195 +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)))))