Average Error: 58.4 → 27.1
Time: 3.1m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot e^{\log \left(\frac{\frac{c0}{h}}{w}\right) + \log \left(\frac{d}{D} \cdot \frac{d}{D}\right)} - M \cdot M} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt{\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right) \cdot \frac{c0}{2 \cdot w} \le 6.198358977764311 \cdot 10^{+276}:\\ \;\;\;\;\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}\right)\right) \cdot \frac{c0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (* (/ c0 (* w 2)) (* (sqrt (+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (* M M))))) (sqrt (+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (exp (+ (log (/ (/ c0 h) w)) (log (* (/ d D) (/ d D)))))) (* M M))))))) < 6.198358977764311e+276

    1. Initial program 47.1

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

    2. Initial simplification22.3

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

    4. Applied associate-*r*22.1

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

    if 6.198358977764311e+276 < (* (/ c0 (* w 2)) (* (sqrt (+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) (* M M))))) (sqrt (+ (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (sqrt (- (* (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) (exp (+ (log (/ (/ c0 h) w)) (log (* (/ d D) (/ d D)))))) (* M M)))))))

    1. Initial program 60.3

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

    2. Initial simplification58.8

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

    3. Taylor expanded around inf 32.6

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \color{blue}{0}\]
    4. Using strategy rm

    5. Applied mul027.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot e^{\log \left(\frac{\frac{c0}{h}}{w}\right) + \log \left(\frac{d}{D} \cdot \frac{d}{D}\right)} - M \cdot M} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt{\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right) \cdot \frac{c0}{2 \cdot w} \le 6.198358977764311 \cdot 10^{+276}:\\ \;\;\;\;\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}\right)\right) \cdot \frac{c0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018215 
(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))))))