Average Error: 58.5 → 54.9
Time: 6.2m
Precision: 64
Internal Precision: 6976
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \le 1.3109132541065102 \cdot 10^{-65}:\\ \;\;\;\;\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{d}^{2} \cdot c0}{\left(w \cdot h\right) \cdot {D}^{2}} \cdot 2\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

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 M < 1.3109132541065102e-65

    1. Initial program 57.6

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

    2. Initial simplification51.3

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

    4. Applied associate-*l/53.1

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

    if 1.3109132541065102e-65 < M

    1. Initial program 61.7

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

    2. Initial simplification60.0

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

    4. Applied associate-*r*60.1

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

    5. Taylor expanded around 0 60.6

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

  4. Final simplification54.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \le 1.3109132541065102 \cdot 10^{-65}:\\ \;\;\;\;\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{h}}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{d}^{2} \cdot c0}{\left(w \cdot h\right) \cdot {D}^{2}} \cdot 2\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2018251 
(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))))))