Average Error: 58.7 → 53.3
Time: 9.0m
Precision: 64
Internal Precision: 6976
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} - M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} + M\right)} + \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}} \cdot \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}}\right) \le 7.369575073012989 \cdot 10^{+289}:\\ \;\;\;\;\left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} - M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} + M\right)} + \left(\sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}} \cdot \left(\sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}} \cdot \sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}}\right)\right) \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + \sqrt{\left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + M\right) \cdot \left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} - M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

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 (* (/ (/ c0 2) w) (+ (sqrt (* (+ M (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) M))) (* (sqrt (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) (sqrt (* (/ (/ c0 w) h) (* (/ d D) (/ d D))))))) < 7.369575073012989e+289

    1. Initial program 49.0

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

    2. Initial simplification23.2

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

    4. Applied associate-*r*23.4

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

    6. Applied add-cube-cbrt23.6

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

    if 7.369575073012989e+289 < (* (/ (/ c0 2) w) (+ (sqrt (* (+ M (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) (- (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) M))) (* (sqrt (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) (sqrt (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))))))

    1. Initial program 60.4

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

    2. Initial simplification58.9

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

    4. Applied add-log-exp62.0

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

    5. Applied add-log-exp61.1

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

    6. Applied sum-log60.9

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

    7. Simplified58.7

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

  4. Final simplification53.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} - M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} + M\right)} + \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}} \cdot \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}}\right) \le 7.369575073012989 \cdot 10^{+289}:\\ \;\;\;\;\left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} - M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h} + M\right)} + \left(\sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}} \cdot \left(\sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}} \cdot \sqrt[3]{\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}}\right)\right) \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + \sqrt{\left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + M\right) \cdot \left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} - M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 9.0m)Debug logProfile

herbie shell --seed 2018217 
(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))))))