Average Error: 58.4 → 28.4
Time: 5.6m
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{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le -7.997073933475495 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{\frac{w \cdot 2}{c0 \cdot M} \cdot \left(\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(M \cdot \left(-M\right)\right))_*}\right)}{M}}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le -2.182835902747342 \cdot 10^{-178}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le 1.6255746857262004 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{\frac{w \cdot 2}{c0 \cdot M} \cdot \left(\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(M \cdot \left(-M\right)\right))_*}\right)}{M}}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le 2.6152844399714374 \cdot 10^{+290}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\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(-M \cdot M\right))_*} + \frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}}\\ \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 (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (- (* M M)))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) 3))) < -7.997073933475495e+104 or -2.182835902747342e-178 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (- (* M M)))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) 3))) < 1.6255746857262004e-85

    1. Initial program 51.3

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

    3. Applied flip-+57.2

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

    4. Applied frac-times57.4

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

    5. Applied simplify51.0

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

    7. Applied clear-num51.1

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

    8. Applied simplify35.0

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

    10. Applied associate-*l/34.0

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

    if -7.997073933475495e+104 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (- (* M M)))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) 3))) < -2.182835902747342e-178 or 1.6255746857262004e-85 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (- (* M M)))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) 3))) < 2.6152844399714374e+290

    1. Initial program 43.9

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

    3. Applied add-cbrt-cube44.1

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

    4. Applied simplify1.9

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

    if 2.6152844399714374e+290 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (fma (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D))) (- (* M M)))) (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))) 3)))

    1. Initial program 61.6

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

    2. Taylor expanded around inf 33.4

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

    3. Applied simplify28.1

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

Runtime

Time bar (total: 5.6m)Debug logProfile

herbie shell --seed 2018170 +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))))))