Average Error: 58.2 → 31.5
Time: 4.7m
Precision: 64
Internal Precision: 7232
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.0176052264210883 \cdot 10^{-147} \lor \neg \left(d \le 1.7267110155684546 \cdot 10^{-141}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \left(\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{1}{w}\right)\\ \end{array}\]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if d < -1.0176052264210883e-147 or 1.7267110155684546e-141 < d

    1. Initial program 57.8

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification54.5

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]

    3. Taylor expanded around inf 30.0

      \[\leadsto \color{blue}{0}\]

    if -1.0176052264210883e-147 < d < 1.7267110155684546e-141

    1. Initial program 61.1

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification42.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
    3. Using strategy rm

    4. Applied div-inv42.8

      \[\leadsto \color{blue}{\left(\frac{c0}{2} \cdot \frac{1}{w}\right)} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]

    5. Applied associate-*l*43.3

      \[\leadsto \color{blue}{\frac{c0}{2} \cdot \left(\frac{1}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt43.9

      \[\leadsto \frac{c0}{2} \cdot \left(\frac{1}{w} \cdot \color{blue}{\left(\sqrt{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.0176052264210883 \cdot 10^{-147} \lor \neg \left(d \le 1.7267110155684546 \cdot 10^{-141}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \left(\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{1}{w}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.631.523.18.51.4%
herbie shell --seed 2018290 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))