Average Error: 58.2 → 33.5
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -7.399400230320045 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 3.1862491261097584 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt{\left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right) \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right) - M \cdot M} + c0 \cdot \left(\frac{1}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 < -7.399400230320045e-257 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. Simplified54.0

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

    3. Taylor expanded around -inf 33.7

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

    4. Taylor expanded around 0 32.0

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

    if -7.399400230320045e-257 < 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. Simplified44.9

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

    4. Applied div-inv46.0

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

    5. Applied associate-*l*48.2

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

  4. Final simplification33.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -7.399400230320045 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 3.1862491261097584 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt{\left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right) \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right) - M \cdot M} + c0 \cdot \left(\frac{1}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(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))))))