Average Error: 58.3 → 54.2
Time: 6.5m
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}\;D \le -3.0023335278674712 \cdot 10^{+69}:\\ \;\;\;\;\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le -1.797111844112005 \cdot 10^{-162}:\\ \;\;\;\;\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 3.4469699345637277 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le 2.576912736730397 \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{else}:\\ \;\;\;\;\left(\sqrt{\left(M + \frac{c0}{h} \cdot \left(\frac{1}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) \cdot \left(\frac{c0}{h} \cdot \left(\frac{1}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M\right)} + \frac{d}{D} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right)\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 < -3.0023335278674712e+69 or -1.797111844112005e-162 < D < 3.4469699345637277e-65

    1. Initial program 61.1

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

    2. Initial simplification54.8

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

    4. Applied add-sqr-sqrt55.1

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

    if -3.0023335278674712e+69 < D < -1.797111844112005e-162 or 3.4469699345637277e-65 < D < 2.576912736730397e+136

    1. Initial program 53.6

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

    2. Initial simplification51.9

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

    4. Applied div-inv52.3

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

    5. Applied associate-*l*53.2

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

    7. Applied div-inv53.5

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

    8. Applied associate-*l*53.8

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

    9. Taylor expanded around 0 53.9

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

    if 2.576912736730397e+136 < D

    1. Initial program 60.9

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

    2. Initial simplification43.9

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

    4. Applied div-inv44.5

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

    5. Applied associate-*l*46.5

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

    7. Applied div-inv46.9

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

    8. Applied associate-*l*47.1

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

    10. Applied associate-*r*47.6

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

  4. Final simplification54.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -3.0023335278674712 \cdot 10^{+69}:\\ \;\;\;\;\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le -1.797111844112005 \cdot 10^{-162}:\\ \;\;\;\;\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 3.4469699345637277 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le 2.576912736730397 \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{else}:\\ \;\;\;\;\left(\sqrt{\left(M + \frac{c0}{h} \cdot \left(\frac{1}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) \cdot \left(\frac{c0}{h} \cdot \left(\frac{1}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M\right)} + \frac{d}{D} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right)\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 6.5m)Debug logProfile

herbie shell --seed 2018227 
(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))))))