Henrywood and Agarwal, Equation (9a) Percentage Accurate: 81.0% → 87.5%
Time: 16.2s
Alternatives: 13
Speedup: TODO×
Unsound rule application detected in e-graph. Results may not be sound. (more) Specification ? \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Enter valid numbers for all inputs
Local Percentage Accuracy vs ?
The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples. Accuracy vs Speed? The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs. Alternative 1? \[\begin{array}{l}
t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\
\mathbf{if}\;t_0 \leq 5 \cdot 10^{+299}:\\
\;\;\;\;w0 \cdot \sqrt{t_0}\\
\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\
\end{array}
\]
Derivation Split input into 2 regimes if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.0000000000000003e299 Initial program 99.4%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
if 5.0000000000000003e299 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) Initial program 45.6%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac46.6%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified46.6%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Step-by-step derivation associate-*r/60.6%
\[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}}
\]
div-inv60.6%
\[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}
\]
metadata-eval60.6%
\[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}
\]
Applied egg-rr 60.6%
\[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}}
\]
Recombined 2 regimes into one program. Final simplification84.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+299}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\
\end{array}
\]
Alternative 2? \[\begin{array}{l}
t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\
\mathbf{if}\;t_0 \leq 5 \cdot 10^{+299}:\\
\;\;\;\;w0 \cdot \sqrt{t_0}\\
\mathbf{else}:\\
\;\;\;\;w0 + w0 \cdot \left(\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}} \cdot -0.125\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.0000000000000003e299 Initial program 99.4%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
if 5.0000000000000003e299 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) Initial program 45.6%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac46.6%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified46.6%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 48.2%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/48.2%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative48.2%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/48.2%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac47.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow247.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative47.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow247.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow247.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified47.2%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity47.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac49.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 49.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity49.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*52.7%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified52.7%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation distribute-rgt-in52.7%
\[\leadsto \color{blue}{1 \cdot w0 + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0}
\]
*-un-lft-identity52.7%
\[\leadsto \color{blue}{w0} + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0
\]
*-commutative52.7%
\[\leadsto w0 + \color{blue}{\left(\left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)} \cdot w0
\]
associate-/l*54.1%
\[\leadsto w0 + \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right) \cdot w0
\]
associate-*l/55.3%
\[\leadsto w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}\right) \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 55.3%
\[\leadsto \color{blue}{w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right) \cdot -0.125\right) \cdot w0}
\]
Step-by-step derivation associate-*l/57.5%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}}} \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 57.5%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}}} \cdot -0.125\right) \cdot w0
\]
Recombined 2 regimes into one program. Final simplification83.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+299}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;w0 + w0 \cdot \left(\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}} \cdot -0.125\right)\\
\end{array}
\]
Alternative 3? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{D}}\right)\\
\mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-262}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 h l) < -inf.0 Initial program 40.0%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac40.0%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified40.0%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 57.5%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/57.5%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative57.5%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/57.5%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac53.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow253.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative53.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow253.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow253.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified53.4%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity53.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac58.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 58.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity58.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*63.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified63.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation distribute-rgt-in63.2%
\[\leadsto \color{blue}{1 \cdot w0 + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0}
\]
*-un-lft-identity63.2%
\[\leadsto \color{blue}{w0} + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0
\]
*-commutative63.2%
\[\leadsto w0 + \color{blue}{\left(\left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)} \cdot w0
\]
associate-/l*68.6%
\[\leadsto w0 + \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right) \cdot w0
\]
associate-*l/68.8%
\[\leadsto w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}\right) \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 68.8%
\[\leadsto \color{blue}{w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right) \cdot -0.125\right) \cdot w0}
\]
Step-by-step derivation frac-times68.8%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{M}}} \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 68.8%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{M}}} \cdot -0.125\right) \cdot w0
\]
if -inf.0 < (/.f64 h l) < -1.00000000000000001e-262 Initial program 80.3%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac78.7%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified78.7%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
if -1.00000000000000001e-262 < (/.f64 h l) Initial program 85.5%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac84.6%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified84.6%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 90.6%
\[\leadsto \color{blue}{w0}
\]
Recombined 3 regimes into one program. Final simplification83.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{D}}\right)\\
\mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-262}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 4? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-130}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -2.0000000000000002e-130 Initial program 73.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac71.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified71.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Step-by-step derivation associate-*r/77.2%
\[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}}
\]
div-inv77.2%
\[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}
\]
metadata-eval77.2%
\[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}
\]
Applied egg-rr 77.2%
\[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}}
\]
Taylor expanded in M around 0 49.2%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation *-commutative49.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell \cdot {d}^{2}}\right)
\]
unpow249.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)
\]
*-commutative49.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right)
\]
unpow249.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r*50.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{\ell \cdot {d}^{2}}\right)
\]
associate-*l*57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}}{\ell \cdot {d}^{2}}\right)
\]
unpow257.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right)
\]
Simplified57.9%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)}
\]
Step-by-step derivation times-frac55.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d}\right)}\right)
\]
Applied egg-rr 55.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d}\right)}\right)
\]
if -2.0000000000000002e-130 < (/.f64 h l) Initial program 85.0%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac84.3%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified84.3%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 86.4%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification71.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-130}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 5? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-187}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -2e-187 Initial program 74.3%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac72.8%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified72.8%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 50.8%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/50.8%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative50.8%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/50.8%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified51.6%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*57.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified57.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
if -2e-187 < (/.f64 h l) Initial program 84.3%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac83.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified83.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 87.3%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification72.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-187}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 6? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-187}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}} \cdot \left(D \cdot D\right)}{\ell}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -2e-187 Initial program 74.3%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac72.8%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified72.8%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 50.8%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/50.8%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative50.8%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/50.8%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow251.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified51.6%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity54.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*57.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified57.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation associate-*l/58.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}{\ell}}\right)
\]
associate-*l/59.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}}{\ell}\right)
\]
Applied egg-rr 59.8%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\ell}}\right)
\]
if -2e-187 < (/.f64 h l) Initial program 84.3%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac83.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified83.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 87.3%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification73.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-187}:\\
\;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}} \cdot \left(D \cdot D\right)}{\ell}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 7? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{M \cdot \frac{h}{d}}{\frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -0.0 Initial program 73.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac71.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified71.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 50.4%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative50.4%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified51.1%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation distribute-rgt-in57.9%
\[\leadsto \color{blue}{1 \cdot w0 + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0}
\]
*-un-lft-identity57.9%
\[\leadsto \color{blue}{w0} + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0
\]
*-commutative57.9%
\[\leadsto w0 + \color{blue}{\left(\left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)} \cdot w0
\]
associate-/l*60.8%
\[\leadsto w0 + \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right) \cdot w0
\]
associate-*l/62.5%
\[\leadsto w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}\right) \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 62.5%
\[\leadsto \color{blue}{w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right) \cdot -0.125\right) \cdot w0}
\]
if -0.0 < (/.f64 h l) Initial program 94.5%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac94.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified94.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 100.0%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification73.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{M \cdot \frac{h}{d}}{\frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 8? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{D}}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -0.0 Initial program 73.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac71.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified71.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 50.4%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative50.4%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified51.1%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation distribute-rgt-in57.9%
\[\leadsto \color{blue}{1 \cdot w0 + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0}
\]
*-un-lft-identity57.9%
\[\leadsto \color{blue}{w0} + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0
\]
*-commutative57.9%
\[\leadsto w0 + \color{blue}{\left(\left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)} \cdot w0
\]
associate-/l*60.8%
\[\leadsto w0 + \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right) \cdot w0
\]
associate-*l/62.5%
\[\leadsto w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}\right) \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 62.5%
\[\leadsto \color{blue}{w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right) \cdot -0.125\right) \cdot w0}
\]
Step-by-step derivation frac-times67.7%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{M}}} \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 67.7%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{M}}} \cdot -0.125\right) \cdot w0
\]
if -0.0 < (/.f64 h l) Initial program 94.5%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac94.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified94.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 100.0%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification76.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{D}}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 9? \[\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 h l) < -0.0 Initial program 73.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac71.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified71.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 50.4%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative50.4%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/50.4%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow251.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified51.1%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity51.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity54.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified57.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Step-by-step derivation distribute-rgt-in57.9%
\[\leadsto \color{blue}{1 \cdot w0 + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0}
\]
*-un-lft-identity57.9%
\[\leadsto \color{blue}{w0} + \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot w0
\]
*-commutative57.9%
\[\leadsto w0 + \color{blue}{\left(\left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right)} \cdot w0
\]
associate-/l*60.8%
\[\leadsto w0 + \left(\left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right) \cdot -0.125\right) \cdot w0
\]
associate-*l/62.5%
\[\leadsto w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}\right) \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 62.5%
\[\leadsto \color{blue}{w0 + \left(\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right) \cdot -0.125\right) \cdot w0}
\]
Step-by-step derivation associate-*l/66.5%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}}} \cdot -0.125\right) \cdot w0
\]
Applied egg-rr 66.5%
\[\leadsto w0 + \left(\color{blue}{\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}}} \cdot -0.125\right) \cdot w0
\]
if -0.0 < (/.f64 h l) Initial program 94.5%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac94.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified94.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 100.0%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification76.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq 0:\\
\;\;\;\;w0 + w0 \cdot \left(\frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\frac{\ell}{D}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 10? \[\begin{array}{l}
\mathbf{if}\;D \leq -2.7 \cdot 10^{+29} \lor \neg \left(D \leq 4.8 \cdot 10^{+243}\right):\\
\;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if D < -2.7e29 or 4.8000000000000001e243 < D Initial program 79.6%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac75.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified75.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 37.3%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/37.3%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative37.3%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/37.3%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac37.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow237.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative37.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow237.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow237.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified37.5%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity37.5%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac39.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 39.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity39.1%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*42.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified42.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Taylor expanded in D around inf 23.3%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}
\]
Step-by-step derivation associate-*r/23.3%
\[\leadsto \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{{d}^{2} \cdot \ell}}
\]
associate-*r*23.3%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot w0\right) \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}
\]
*-commutative23.3%
\[\leadsto \frac{-0.125 \cdot \left(\left({D}^{2} \cdot w0\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2} \cdot \ell}
\]
associate-*r*23.3%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right)}}{{d}^{2} \cdot \ell}
\]
associate-*r/23.3%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot \ell}}
\]
times-frac23.6%
\[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)}
\]
unpow223.6%
\[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
unpow223.6%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
associate-/l*23.6%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \color{blue}{\frac{w0}{\frac{\ell}{h \cdot {M}^{2}}}}\right)
\]
unpow223.6%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right)
\]
Simplified23.6%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)}
\]
Step-by-step derivation times-frac26.1%
\[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)
\]
Applied egg-rr 26.1%
\[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)
\]
if -2.7e29 < D < 4.8000000000000001e243 Initial program 79.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac79.1%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified79.1%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 70.7%
\[\leadsto \color{blue}{w0}
\]
Recombined 2 regimes into one program. Final simplification58.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;D \leq -2.7 \cdot 10^{+29} \lor \neg \left(D \leq 4.8 \cdot 10^{+243}\right):\\
\;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 11? \[\begin{array}{l}
\mathbf{if}\;D \leq -2.7 \cdot 10^{+29}:\\
\;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\frac{\ell}{M \cdot h}}{M}}\right)\\
\mathbf{elif}\;D \leq 5.7 \cdot 10^{+243}:\\
\;\;\;\;w0\\
\mathbf{else}:\\
\;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if D < -2.7e29 Initial program 78.8%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac75.6%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified75.6%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 37.9%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/37.9%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative37.9%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/37.9%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac38.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow238.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative38.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow238.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow238.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified38.0%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity38.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac40.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 40.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity40.0%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*43.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified43.4%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Taylor expanded in D around inf 21.2%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}
\]
Step-by-step derivation associate-*r/21.2%
\[\leadsto \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{{d}^{2} \cdot \ell}}
\]
associate-*r*21.1%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot w0\right) \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}
\]
*-commutative21.1%
\[\leadsto \frac{-0.125 \cdot \left(\left({D}^{2} \cdot w0\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2} \cdot \ell}
\]
associate-*r*21.2%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right)}}{{d}^{2} \cdot \ell}
\]
associate-*r/21.2%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot \ell}}
\]
times-frac21.3%
\[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)}
\]
unpow221.3%
\[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
unpow221.3%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
associate-/l*21.3%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \color{blue}{\frac{w0}{\frac{\ell}{h \cdot {M}^{2}}}}\right)
\]
unpow221.3%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right)
\]
Simplified21.3%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)}
\]
Taylor expanded in l around 0 21.3%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\color{blue}{\frac{\ell}{h \cdot {M}^{2}}}}\right)
\]
Step-by-step derivation unpow221.3%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right)
\]
associate-*l*24.8%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{\color{blue}{\left(h \cdot M\right) \cdot M}}}\right)
\]
associate-/r*25.0%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\color{blue}{\frac{\frac{\ell}{h \cdot M}}{M}}}\right)
\]
*-commutative25.0%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\frac{\ell}{\color{blue}{M \cdot h}}}{M}}\right)
\]
Simplified25.0%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\color{blue}{\frac{\frac{\ell}{M \cdot h}}{M}}}\right)
\]
if -2.7e29 < D < 5.7000000000000003e243 Initial program 79.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac79.1%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified79.1%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 70.7%
\[\leadsto \color{blue}{w0}
\]
if 5.7000000000000003e243 < D Initial program 83.2%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac75.0%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified75.0%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 34.0%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/34.0%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative34.0%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/34.0%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow234.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow234.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow234.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified34.9%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified34.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Taylor expanded in D around inf 33.7%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}
\]
Step-by-step derivation associate-*r/33.7%
\[\leadsto \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{{d}^{2} \cdot \ell}}
\]
associate-*r*34.0%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot w0\right) \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}
\]
*-commutative34.0%
\[\leadsto \frac{-0.125 \cdot \left(\left({D}^{2} \cdot w0\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2} \cdot \ell}
\]
associate-*r*33.7%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right)}}{{d}^{2} \cdot \ell}
\]
associate-*r/33.7%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot \ell}}
\]
times-frac34.5%
\[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)}
\]
unpow234.5%
\[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
unpow234.5%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
associate-/l*34.5%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \color{blue}{\frac{w0}{\frac{\ell}{h \cdot {M}^{2}}}}\right)
\]
unpow234.5%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right)
\]
Simplified34.5%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)}
\]
Step-by-step derivation times-frac36.3%
\[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)
\]
Applied egg-rr 36.3%
\[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)
\]
Recombined 3 regimes into one program. Final simplification58.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;D \leq -2.7 \cdot 10^{+29}:\\
\;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0}{\frac{\frac{\ell}{M \cdot h}}{M}}\right)\\
\mathbf{elif}\;D \leq 5.7 \cdot 10^{+243}:\\
\;\;\;\;w0\\
\mathbf{else}:\\
\;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0}{\frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\
\end{array}
\]
Alternative 12? \[\begin{array}{l}
\mathbf{if}\;d \leq -8.6 \cdot 10^{-97}:\\
\;\;\;\;w0\\
\mathbf{elif}\;d \leq 1.85 \cdot 10^{-269}:\\
\;\;\;\;\left(-0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot w0\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Derivation Split input into 2 regimes if d < -8.6e-97 or 1.84999999999999989e-269 < d Initial program 82.8%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac81.5%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified81.5%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 73.7%
\[\leadsto \color{blue}{w0}
\]
if -8.6e-97 < d < 1.84999999999999989e-269 Initial program 63.1%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac63.0%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified63.0%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 39.1%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}
\]
Step-by-step derivation associate-*r/39.1%
\[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative39.1%
\[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)
\]
associate-*r/39.1%
\[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right)
\]
times-frac38.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)
\]
unpow238.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)
\]
*-commutative38.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)
\]
unpow238.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)
\]
unpow238.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)
\]
Simplified38.9%
\[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}
\]
Step-by-step derivation *-un-lft-identity38.9%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right)
\]
times-frac43.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(1 \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right)
\]
Applied egg-rr 43.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(1 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)}\right)\right)
\]
Step-by-step derivation *-lft-identity43.2%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)
\]
associate-/l*45.7%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right)\right)\right)
\]
Simplified45.7%
\[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)}\right)\right)
\]
Taylor expanded in D around inf 36.9%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}
\]
Step-by-step derivation associate-*r/36.9%
\[\leadsto \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{{d}^{2} \cdot \ell}}
\]
associate-*r*36.9%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot w0\right) \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}
\]
*-commutative36.9%
\[\leadsto \frac{-0.125 \cdot \left(\left({D}^{2} \cdot w0\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2} \cdot \ell}
\]
associate-*r*36.9%
\[\leadsto \frac{-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right)}}{{d}^{2} \cdot \ell}
\]
associate-*r/36.9%
\[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot \ell}}
\]
times-frac37.2%
\[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)}
\]
unpow237.2%
\[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
unpow237.2%
\[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
times-frac42.4%
\[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
unpow242.4%
\[\leadsto -0.125 \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)
\]
associate-*r*42.4%
\[\leadsto \color{blue}{\left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}}
\]
*-commutative42.4%
\[\leadsto \left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot \frac{w0 \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}
\]
unpow242.4%
\[\leadsto \left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot \frac{w0 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{\ell}
\]
associate-*r*42.5%
\[\leadsto \left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot \frac{w0 \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{\ell}
\]
Simplified44.6%
\[\leadsto \color{blue}{\left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot \left(M \cdot \left(\left(M \cdot w0\right) \cdot \frac{h}{\ell}\right)\right)}
\]
Step-by-step derivation unpow244.6%
\[\leadsto \left(-0.125 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \left(M \cdot \left(\left(M \cdot w0\right) \cdot \frac{h}{\ell}\right)\right)
\]
Applied egg-rr 44.6%
\[\leadsto \left(-0.125 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \left(M \cdot \left(\left(M \cdot w0\right) \cdot \frac{h}{\ell}\right)\right)
\]
Recombined 2 regimes into one program. Final simplification68.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -8.6 \cdot 10^{-97}:\\
\;\;\;\;w0\\
\mathbf{elif}\;d \leq 1.85 \cdot 10^{-269}:\\
\;\;\;\;\left(-0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot w0\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;w0\\
\end{array}
\]
Alternative 13? \[w0
\]
Derivation Initial program 79.2%
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
\]
Step-by-step derivation times-frac78.1%
\[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}
\]
Simplified78.1%
\[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}}
\]
Taylor expanded in M around 0 63.6%
\[\leadsto \color{blue}{w0}
\]
Final simplification63.6%
\[\leadsto w0
\]
Reproduce ? herbie shell --seed 2023166
(FPCore (w0 M D h l d)
:name "Henrywood and Agarwal, Equation (9a)"
:precision binary64
(* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
