Average Error: 58.2 → 33.3
Time: 3.2m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.409980931022629 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 2.4001768924480664 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 2 regimes

  2. if d < -1.409980931022629e-158 or 2.4001768924480664e-72 < d

    1. Initial program 58.2

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

    2. Simplified54.0

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

    4. Applied associate-*r*54.8

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

    6. Applied *-un-lft-identity54.8

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

    7. Applied *-un-lft-identity54.8

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

    8. Applied distribute-lft-out54.8

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

    9. Simplified51.8

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

    10. Taylor expanded around inf 31.7

      \[\leadsto \frac{\color{blue}{0}}{2}\]

    if -1.409980931022629e-158 < d < 2.4001768924480664e-72

    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. Simplified41.7

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

    4. Applied associate-*r*43.9

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

    6. Applied *-un-lft-identity43.9

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

    7. Applied *-un-lft-identity43.9

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

    8. Applied distribute-lft-out43.9

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

    9. Simplified41.9

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

    11. Applied associate-*r/42.0

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

  4. Final simplification33.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.409980931022629 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 2.4001768924480664 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + \sqrt{\left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{h} \cdot \frac{\frac{d}{D}}{w} - M\right)}\right)}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(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))))))