Average Error: 58.2 → 33.0
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.1321402208723985 \cdot 10^{-255}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 3.1862491261097584 \cdot 10^{-113}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot \sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}} \cdot \sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 2 regimes

  2. if d < -2.1321402208723985e-255 or 3.1862491261097584e-113 < d

    1. Initial program 58.0

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

    2. Simplified53.0

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

    4. Applied associate-*r*53.8

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

    6. Applied *-un-lft-identity53.8

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

    7. Applied *-un-lft-identity53.8

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

    8. Applied distribute-lft-out53.8

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

    9. Simplified50.8

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

    10. Taylor expanded around inf 32.0

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

    if -2.1321402208723985e-255 < d < 3.1862491261097584e-113

    1. Initial program 59.9

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

    2. Simplified43.1

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

    4. Applied associate-*r*44.4

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

    6. Applied *-un-lft-identity44.4

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

    7. Applied *-un-lft-identity44.4

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

    8. Applied distribute-lft-out44.4

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

    9. Simplified42.9

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

    11. Applied add-cube-cbrt43.1

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}} \cdot \sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}}\right) \cdot \sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}}\right)} \cdot \frac{c0}{w}\]

    12. Applied associate-*l*43.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}} \cdot \sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}}\right) \cdot \left(\sqrt[3]{\frac{1 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(M + \frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{2}} \cdot \frac{c0}{w}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification33.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.1321402208723985 \cdot 10^{-255}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 3.1862491261097584 \cdot 10^{-113}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot \sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}} \cdot \sqrt[3]{\frac{\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} - M\right) \cdot \left(\frac{c0 \cdot \frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{w} + M\right)}}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(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))))))