Average Error: 58.5 → 52.8
Time: 5.4m
Precision: 64
Internal Precision: 6720
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\left(\sqrt{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}\right) - M\right) \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}\right) + M\right)} + \frac{\frac{d}{D} \cdot \frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\]

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. Initial program 58.5

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

  2. Initial simplification53.5

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

  4. Applied associate-*r*54.3

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

  6. Applied associate-*r*53.9

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

  8. Applied associate-*r*51.7

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

  10. Applied associate-*l/52.8

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

  11. Final simplification52.8

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

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018234 
(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))))))