Average Error: 58.1 → 55.9
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;h \le 4.6495075590286495 \cdot 10^{-125}:\\ \;\;\;\;\log \left(e^{\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) + M\right) \cdot \left(\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) - M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{\frac{c0}{h}}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{D} - M\right)} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

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 h < 4.6495075590286495e-125

    1. Initial program 58.4

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

    2. Initial simplification54.0

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

    4. Applied associate-*l/55.2

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

    6. Applied add-log-exp60.0

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

    7. Applied add-log-exp59.6

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

    8. Applied sum-log59.5

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

    9. Simplified56.7

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

    if 4.6495075590286495e-125 < h

    1. Initial program 57.2

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

    2. Initial simplification51.7

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

    4. Applied associate-*r*52.9

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

    6. Applied associate-*r/53.9

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

    7. Applied associate-*r/54.3

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

  4. Final simplification55.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;h \le 4.6495075590286495 \cdot 10^{-125}:\\ \;\;\;\;\log \left(e^{\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) + M\right) \cdot \left(\frac{\frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{d}{D}\right) - M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{\frac{c0}{h}}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{D} - M\right)} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 5.2m)Debug logProfile

herbie shell --seed 2018285 
(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))))))