Average Error: 58.4 → 27.3
Time: 5.3m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right) \le -2.660503551118866 \cdot 10^{+298} \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right) \le 2.0135037245470548 \cdot 10^{+286}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}}\right) \cdot \frac{c0}{2 \cdot w}\\ \end{array}\]

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 (* (/ c0 (* w 2)) (+ (* (/ d D) (* (/ d D) (/ c0 (* h w)))) (sqrt (- (* (* (* (/ d D) (/ d D)) (/ c0 (* h w))) (* (* (/ d D) (/ d D)) (/ c0 (* h w)))) (* M M))))) < -2.660503551118866e+298 or 2.0135037245470548e+286 < (* (/ c0 (* w 2)) (+ (* (/ d D) (* (/ d D) (/ c0 (* h w)))) (sqrt (- (* (* (* (/ d D) (/ d D)) (/ c0 (* h w))) (* (* (/ d D) (/ d D)) (/ c0 (* h w)))) (* M M)))))

    1. Initial program 61.8

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

    2. Initial simplification62.5

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

    3. Taylor expanded around inf 34.6

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

    5. Applied mul029.6

      \[\leadsto \color{blue}{0}\]

    if -2.660503551118866e+298 < (* (/ c0 (* w 2)) (+ (* (/ d D) (* (/ d D) (/ c0 (* h w)))) (sqrt (- (* (* (* (/ d D) (/ d D)) (/ c0 (* h w))) (* (* (/ d D) (/ d D)) (/ c0 (* h w)))) (* M M))))) < 2.0135037245470548e+286

    1. Initial program 44.8

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

    2. Initial simplification17.8

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

    4. Applied add-cube-cbrt18.0

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

  4. Final simplification27.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right) \le -2.660503551118866 \cdot 10^{+298} \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right) \le 2.0135037245470548 \cdot 10^{+286}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}}\right) \cdot \frac{c0}{2 \cdot w}\\ \end{array}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed 2018214 
(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))))))