Average Error: 58.4 → 27.1
Time: 3.1m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{\frac{c0}{w}}{h}\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right))_*}{\frac{2}{\frac{c0}{w}}} \le -2.660503551118866 \cdot 10^{+298}:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{(\left(\frac{\frac{c0}{w}}{h}\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right))_*}{\frac{2}{\frac{c0}{w}}} \le 2.0135037245470548 \cdot 10^{+286}:\\ \;\;\;\;\frac{\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \frac{d}{D}\right) + \left(\left(-M\right) \cdot M\right))_*} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}}{\frac{2}{\frac{c0}{w}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (/ (fma (/ (/ c0 w) h) (* (/ d D) (/ d D)) (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (- M) M)))) (/ 2 (/ c0 w))) < -2.660503551118866e+298 or 2.0135037245470548e+286 < (/ (fma (/ (/ c0 w) h) (* (/ d D) (/ d D)) (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (- M) M)))) (/ 2 (/ c0 w)))

    1. Initial program 61.7

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

    2. Initial simplification62.4

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

    3. Taylor expanded around inf 34.2

      \[\leadsto \frac{\color{blue}{0}}{\frac{2}{\frac{c0}{w}}}\]
    4. Using strategy rm

    5. Applied div029.2

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

    if -2.660503551118866e+298 < (/ (fma (/ (/ c0 w) h) (* (/ d D) (/ d D)) (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (- M) M)))) (/ 2 (/ c0 w))) < 2.0135037245470548e+286

    1. Initial program 46.1

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

    2. Initial simplification17.1

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

    4. Applied associate-*r*19.5

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{\frac{c0}{w}}{h}\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right))_*}{\frac{2}{\frac{c0}{w}}} \le -2.660503551118866 \cdot 10^{+298}:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{(\left(\frac{\frac{c0}{w}}{h}\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right))_*}{\frac{2}{\frac{c0}{w}}} \le 2.0135037245470548 \cdot 10^{+286}:\\ \;\;\;\;\frac{\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \frac{d}{D}\right) + \left(\left(-M\right) \cdot M\right))_*} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}}{\frac{2}{\frac{c0}{w}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018214 +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))))))