Average Error: 57.7 → 49.6
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\frac{\sqrt{\left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D} \cdot c0}{h} - M\right) \cdot \left(M + \frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D} \cdot c0}{h}\right)} + \frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D} \cdot c0}{h}}{2} \cdot \frac{c0}{w}\]

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

Derivation

  1. Initial program 57.7

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

  2. Simplified51.8

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

  4. Applied associate-*r*52.7

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

  6. Applied *-un-lft-identity52.7

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

  7. Applied *-un-lft-identity52.7

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

  8. Applied distribute-lft-out52.7

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

  9. Simplified49.6

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

  10. Final simplification49.6

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

Reproduce

herbie shell --seed 2019089 
(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))))))