Average Error: 57.8 → 55.3
Time: 7.4m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;D \le -3.0365187498466197 \cdot 10^{-31}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} - M\right)} + \left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le -1.448081804081321 \cdot 10^{-136}:\\ \;\;\;\;\left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(h \cdot w\right) \cdot {D}^{2}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le 6.4228499417539 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M} \cdot \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} - M\right)} + \left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if D < -3.0365187498466197e-31 or 6.4228499417539e-200 < D

    1. Initial program 56.3

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

    2. Initial simplification50.5

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

    4. Applied associate-*r*51.7

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

    6. Applied add-sqr-sqrt52.4

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

    8. Applied add-log-exp58.9

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

    9. Applied add-log-exp58.0

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

    10. Applied sum-log57.9

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

    11. Simplified54.4

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

    if -3.0365187498466197e-31 < D < -1.448081804081321e-136

    1. Initial program 52.3

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

    2. Initial simplification52.5

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

    4. Applied associate-*r*53.5

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

    6. Applied add-sqr-sqrt54.2

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

    7. Taylor expanded around 0 52.7

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

    if -1.448081804081321e-136 < D < 6.4228499417539e-200

    1. Initial program 62.3

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

    2. Initial simplification56.2

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

    4. Applied associate-*r*56.8

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

    6. Applied sqrt-prod57.7

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

  4. Final simplification55.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -3.0365187498466197 \cdot 10^{-31}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} - M\right)} + \left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le -1.448081804081321 \cdot 10^{-136}:\\ \;\;\;\;\left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(h \cdot w\right) \cdot {D}^{2}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le 6.4228499417539 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M} \cdot \sqrt{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h} - M\right)} + \left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{d}{D}}{h}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 7.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes55.655.347.08.74.2%
herbie shell --seed 2018353 
(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))))))