Average Error: 58.4 → 28.9
Time: 5.7m
Precision: 64
Internal Precision: 7232
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le -8.113568444663876 \cdot 10^{-34}:\\ \;\;\;\;c0 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) - M \cdot M}}{w \cdot 2}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le 1.508955565548636 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{\left(\frac{2}{c0 \cdot M} \cdot \frac{w}{M}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) - M \cdot M}\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le +\infty:\\ \;\;\;\;c0 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) - M \cdot M}}{w \cdot 2}\\ \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

Derivation

  1. Split input into 3 regimes

  2. if (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3))) < -8.113568444663876e-34 or 1.508955565548636e-133 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3))) < +inf.0

    1. Initial program 55.4

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

    3. Applied div-inv55.4

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

    4. Applied associate-*l*55.5

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

    5. Applied simplify46.8

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

    if -8.113568444663876e-34 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3))) < 1.508955565548636e-133

    1. Initial program 47.0

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

    3. Applied flip-+54.1

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

    4. Applied frac-times54.3

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

    5. Applied simplify48.3

      \[\leadsto \frac{\color{blue}{\left(0 + M \cdot M\right) \cdot c0}}{\left(2 \cdot w\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\]
    6. Using strategy rm

    7. Applied clear-num48.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(2 \cdot w\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\left(0 + M \cdot M\right) \cdot c0}}}\]

    8. Applied simplify29.6

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

    10. Applied *-un-lft-identity29.6

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

    11. Applied associate-*r*29.6

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

    12. Applied simplify27.1

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

    if +inf.0 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3)))

    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. Taylor expanded around inf 30.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]

    3. Applied simplify24.1

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

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))))))