Henrywood and Agarwal, Equation (12) Percentage Accurate: 67.6% → 71.5%
Time: 24.5s
Alternatives: 21
Speedup: 3.2×
Unsound rule application detected in e-graph. Results may not be sound. (more) Specification ? \[\begin{array}{l}
t_0 := \frac{1}{2}\\
\left({\left(\frac{d}{h}\right)}^{t_0} \cdot {\left(\frac{d}{\ell}\right)}^{t_0}\right) \cdot \left(1 - \left(t_0 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\end{array}
\]
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: 71.5% accurate, 0.6× speedup? \[\begin{array}{l}
t_0 := D \cdot \frac{M}{d \cdot 2}\\
t_1 := \sqrt{\frac{d}{h}}\\
t_2 := \sqrt{\frac{d}{\ell}}\\
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;t_1 \cdot \left(t_2 \cdot \left(1 - 0.5 \cdot {\left(t_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\
\mathbf{elif}\;d \leq 1.05 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;t_1 \cdot \left(t_2 \cdot \left(1 - 0.5 \cdot {\left(t_0 \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if d < -1.999999999999994e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified71.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Step-by-step derivation associate-*r/69.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
frac-times70.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative70.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative70.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
associate-*r/69.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
associate-/r*69.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
Applied egg-rr 69.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
Step-by-step derivation *-commutative69.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}}{\ell}\right)\right)
\]
associate-/l/69.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}}{\ell}\right)\right)
\]
associate-*l/71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)}\right)\right)
\]
add-sqr-sqrt71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}\right)\right)
\]
pow271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}^{2}}\right)\right)
\]
Applied egg-rr 73.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right)
\]
if -1.999999999999994e-310 < d < 1.05e-197 Initial program 27.8%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 43.2%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative43.2%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*38.2%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*37.5%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.5%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified37.5%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow243.2%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/37.5%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative38.2%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
if 1.05e-197 < d Initial program 79.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*78.9%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval78.9%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified78.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Step-by-step derivation associate-*r/80.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
frac-times81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
associate-*r/81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
associate-/r*81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
Applied egg-rr 81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
Step-by-step derivation *-commutative81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}}{\ell}\right)\right)
\]
associate-/l/81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}}{\ell}\right)\right)
\]
associate-*l/78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)}\right)\right)
\]
add-sqr-sqrt78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}\right)\right)
\]
pow278.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}^{2}}\right)\right)
\]
Applied egg-rr 81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right)
\]
Step-by-step derivation sqrt-div85.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{\ell}}}\right)}^{2}\right)\right)
\]
Applied egg-rr 85.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{\ell}}}\right)}^{2}\right)\right)
\]
Recombined 3 regimes into one program. Final simplification77.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\
\mathbf{elif}\;d \leq 1.05 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\right)\\
\end{array}
\]
Alternative 2: 71.5% accurate, 0.4× speedup? \[\begin{array}{l}
\mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0 Initial program 85.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*85.3%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval85.3%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/285.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval85.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/285.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*85.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval85.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac84.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified84.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Step-by-step derivation associate-*r/83.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
frac-times84.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative84.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative84.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
associate-*r/83.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
associate-/r*83.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
Applied egg-rr 83.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
Step-by-step derivation *-commutative83.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}}{\ell}\right)\right)
\]
associate-/l/83.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}}{\ell}\right)\right)
\]
associate-*l/85.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)}\right)\right)
\]
add-sqr-sqrt85.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}\right)\right)
\]
pow285.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\right)}^{2}}\right)\right)
\]
Applied egg-rr 87.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right)
\]
if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) Initial program 0.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 7.9%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative7.9%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*7.9%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*7.7%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/5.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*5.4%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified5.4%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 7.9%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow27.9%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/5.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/5.4%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*5.3%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative5.3%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified5.3%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
Recombined 2 regimes into one program. Final simplification76.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\end{array}
\]
Alternative 3: 71.0% accurate, 0.5× speedup? \[\begin{array}{l}
t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\
\mathbf{if}\;t_0 \leq \infty:\\
\;\;\;\;t_0\\
\mathbf{else}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0 Initial program 85.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) Initial program 0.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 7.9%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative7.9%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*7.9%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*7.7%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/5.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*5.4%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified5.4%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 7.9%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow27.9%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/5.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow25.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/5.4%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*5.3%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative5.3%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified5.3%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow9.7%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified19.2%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
Recombined 2 regimes into one program. Final simplification74.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\
\;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\end{array}
\]
Alternative 4: 69.6% accurate, 1.0× speedup? \[\begin{array}{l}
t_0 := \sqrt{\frac{d}{h}}\\
t_1 := \sqrt{\frac{d}{\ell}}\\
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 5.1 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if d < -1.999999999999994e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified71.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
if -1.999999999999994e-310 < d < 5.1000000000000003e-197 Initial program 27.8%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 43.2%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative43.2%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*38.2%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*37.5%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.5%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified37.5%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow243.2%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/37.5%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative38.2%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
if 5.1000000000000003e-197 < d Initial program 79.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*78.9%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval78.9%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified78.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Step-by-step derivation associate-*r/80.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
frac-times81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
associate-*r/81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
associate-/r*81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
Applied egg-rr 81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
Step-by-step derivation div-inv81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)}\right)\right)
\]
associate-/l/81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left({\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)\right)\right)
\]
*-commutative81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left({\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)\right)\right)
\]
Applied egg-rr 81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)}\right)\right)
\]
Recombined 3 regimes into one program. Final simplification75.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 5.1 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right)\right)\\
\end{array}
\]
Alternative 5: 68.8% accurate, 1.0× speedup? \[\begin{array}{l}
t_0 := \sqrt{\frac{d}{h}}\\
t_1 := \sqrt{\frac{d}{\ell}}\\
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 3 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if d < -1.999999999999994e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified71.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
if -1.999999999999994e-310 < d < 3.00000000000000026e-197 Initial program 27.8%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 43.2%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative43.2%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*38.2%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*37.5%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.5%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified37.5%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow243.2%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/37.5%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative38.2%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
if 3.00000000000000026e-197 < d Initial program 79.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*78.9%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval78.9%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right)
\]
associate-*l*78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right)
\]
*-commutative78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right)
\]
Simplified78.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)}
\]
Step-by-step derivation fma-udef78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)}\right)
\]
associate-*r/78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-*l/78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
frac-times78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
*-commutative78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
*-commutative78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-*r/78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-/r*78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-/r/78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \color{blue}{\left(\frac{-0.5}{\ell} \cdot h\right)} + 1\right)\right)
\]
Applied egg-rr 78.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(\frac{-0.5}{\ell} \cdot h\right) + 1\right)}\right)
\]
Recombined 3 regimes into one program. Final simplification73.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 3 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)\\
\end{array}
\]
Alternative 6: 69.6% accurate, 1.0× speedup? \[\begin{array}{l}
t_0 := \sqrt{\frac{d}{h}}\\
t_1 := \sqrt{\frac{d}{\ell}}\\
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 5.1 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if d < -1.999999999999994e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified71.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
if -1.999999999999994e-310 < d < 5.1000000000000003e-197 Initial program 27.8%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 43.2%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative43.2%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*38.2%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*37.5%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.5%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified37.5%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 43.2%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow243.2%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/37.4%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow237.4%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/37.5%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative38.2%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified38.2%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow48.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified64.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
if 5.1000000000000003e-197 < d Initial program 79.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*78.9%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval78.9%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/278.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval78.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac78.0%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified78.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Step-by-step derivation associate-*r/80.6%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
frac-times81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
*-commutative81.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
associate-*r/81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}\right)\right)
\]
associate-/r*81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\ell}\right)\right)
\]
Applied egg-rr 81.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right)\right)
\]
Recombined 3 regimes into one program. Final simplification75.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\
\mathbf{elif}\;d \leq 5.1 \cdot 10^{-197}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\\
\end{array}
\]
Alternative 7: 67.0% accurate, 1.0× speedup? \[\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)
\]
Derivation Initial program 71.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.3%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.3%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.3%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right)
\]
associate-*l*71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right)
\]
*-commutative71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right)
\]
Simplified70.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)}
\]
Step-by-step derivation fma-udef70.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)}\right)
\]
associate-*r/70.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-*l/70.9%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
frac-times71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
*-commutative71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
*-commutative71.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-*r/71.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-/r*71.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}} + 1\right)\right)
\]
associate-/r/71.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \color{blue}{\left(\frac{-0.5}{\ell} \cdot h\right)} + 1\right)\right)
\]
Applied egg-rr 71.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(\frac{-0.5}{\ell} \cdot h\right) + 1\right)}\right)
\]
Final simplification71.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)
\]
Alternative 8: 68.4% accurate, 1.4× speedup? \[\begin{array}{l}
t_0 := {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\\
t_1 := -0.5 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\\
t_2 := d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\
\mathbf{if}\;\ell \leq -1.55 \cdot 10^{-170}:\\
\;\;\;\;t_2 \cdot \left(-1 - t_1\right)\\
\mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-92}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\
\mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+206}:\\
\;\;\;\;\left(1 + t_1\right) \cdot t_2\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) \cdot 0.25\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if l < -1.54999999999999993e-170 Initial program 68.6%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u37.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef29.1%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 25.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def30.5%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p57.0%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg57.0%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/257.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative57.0%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified57.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around -inf 73.4%
\[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation mul-1-neg73.4%
\[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
distribute-rgt-neg-in73.4%
\[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-173.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square73.4%
\[\leadsto \left(d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt73.2%
\[\leadsto \left(d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr73.2%
\[\leadsto \left(d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt73.4%
\[\leadsto \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified73.4%
\[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if -1.54999999999999993e-170 < l < 2.4000000000000001e-92 Initial program 79.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u22.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef21.4%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 16.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def18.0%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p72.6%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg72.6%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/272.6%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative72.6%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified72.6%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Step-by-step derivation associate-*l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\right)
\]
associate-/l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2}}{\ell}\right)
\]
*-commutative77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{\color{blue}{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}}{\ell}\right)
\]
associate-/l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}\right)
\]
*-commutative77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)
\]
Applied egg-rr 77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right)
\]
if 2.4000000000000001e-92 < l < 3.9e206 Initial program 70.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u42.8%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef31.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 19.6%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def28.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p49.2%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg49.2%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/249.2%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative49.2%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in49.2%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval49.2%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/49.2%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified49.2%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around 0 77.2%
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation *-commutative77.2%
\[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative77.2%
\[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-177.2%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval77.2%
\[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr77.2%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square77.2%
\[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt77.2%
\[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr77.2%
\[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt77.2%
\[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified77.2%
\[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if 3.9e206 < l Initial program 62.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*62.2%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval62.2%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/262.2%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval62.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/262.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
associate-*l*62.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)
\]
metadata-eval62.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
times-frac62.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)
\]
Simplified62.2%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}
\]
Taylor expanded in M around 0 40.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right)
\]
Step-by-step derivation *-commutative40.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25\right)}\right)\right)
\]
*-commutative40.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot 0.25\right)\right)\right)
\]
times-frac49.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot 0.25\right)\right)\right)
\]
unpow249.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
unpow249.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
times-frac53.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
*-commutative53.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
unpow253.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
associate-*r*57.3%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}\right) \cdot 0.25\right)\right)\right)
\]
associate-/l*69.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}\right) \cdot 0.25\right)\right)\right)
\]
Simplified69.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) \cdot 0.25\right)}\right)\right)
\]
Recombined 4 regimes into one program. Final simplification75.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -1.55 \cdot 10^{-170}:\\
\;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\
\mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-92}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\\
\mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+206}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) \cdot 0.25\right)\right)\right)\\
\end{array}
\]
Alternative 9: 70.5% accurate, 1.4× speedup? \[\begin{array}{l}
t_0 := {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\\
t_1 := -0.5 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\\
t_2 := d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\
\mathbf{if}\;\ell \leq -9 \cdot 10^{-173}:\\
\;\;\;\;t_2 \cdot \left(-1 - t_1\right)\\
\mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-93}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\
\mathbf{elif}\;\ell \leq 9 \cdot 10^{+99}:\\
\;\;\;\;\left(1 + t_1\right) \cdot t_2\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if l < -9.00000000000000037e-173 Initial program 68.6%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u37.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef29.1%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 25.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def30.5%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p57.0%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg57.0%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/257.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative57.0%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/57.0%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified57.0%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around -inf 73.4%
\[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation mul-1-neg73.4%
\[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
distribute-rgt-neg-in73.4%
\[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-173.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr73.4%
\[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square73.4%
\[\leadsto \left(d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt73.2%
\[\leadsto \left(d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr73.2%
\[\leadsto \left(d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt73.4%
\[\leadsto \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified73.4%
\[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if -9.00000000000000037e-173 < l < 3.3000000000000001e-93 Initial program 79.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u22.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef21.4%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 16.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def18.0%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p72.6%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg72.6%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/272.6%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative72.6%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/72.6%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified72.6%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Step-by-step derivation associate-*l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\right)
\]
associate-/l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2}}{\ell}\right)
\]
*-commutative77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{\color{blue}{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}}{\ell}\right)
\]
associate-/l/77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}\right)
\]
*-commutative77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)
\]
Applied egg-rr 77.3%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right)
\]
if 3.3000000000000001e-93 < l < 8.9999999999999999e99 Initial program 80.9%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u42.1%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef30.0%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 16.9%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def27.5%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p57.5%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg57.5%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/257.5%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative57.5%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in57.5%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval57.5%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/57.5%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified57.5%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around 0 87.7%
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation *-commutative87.7%
\[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative87.7%
\[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-187.7%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval87.7%
\[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr87.8%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square87.8%
\[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt87.8%
\[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr87.8%
\[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt87.8%
\[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified87.8%
\[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if 8.9999999999999999e99 < l Initial program 55.5%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 47.6%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u47.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/213.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 4 regimes into one program. Final simplification74.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -9 \cdot 10^{-173}:\\
\;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\
\mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-93}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\\
\mathbf{elif}\;\ell \leq 9 \cdot 10^{+99}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 10: 57.1% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+100}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if l < -4.999999999999985e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right)
\]
fma-def71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right)
\]
Simplified71.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)}
\]
Taylor expanded in h around 0 35.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right)
\]
if -4.999999999999985e-310 < l < 1.34999999999999999e100 Initial program 79.7%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u32.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef25.5%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 15.9%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def22.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p64.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg64.4%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/264.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative64.4%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in64.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval64.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/64.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified64.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around 0 80.7%
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation *-commutative80.7%
\[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative80.7%
\[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-180.7%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval80.7%
\[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr80.7%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square80.7%
\[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt80.7%
\[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr80.7%
\[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt80.7%
\[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified80.7%
\[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if 1.34999999999999999e100 < l Initial program 55.5%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 47.6%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u47.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/213.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 3 regimes into one program. Final simplification54.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+100}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 11: 62.2% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;\ell \leq 2.35 \cdot 10^{-301}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(0.125 \cdot \left(M \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\\
\mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+100}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if l < 2.3499999999999998e-301 Initial program 71.5%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in M around 0 37.9%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right)
\]
Step-by-step derivation associate-*r/37.9%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)
\]
*-commutative37.9%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)
\]
associate-*r/37.9%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)
\]
*-commutative37.9%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right)
\]
times-frac41.0%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot 0.125\right)
\]
associate-*l*41.0%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot 0.125\right)}\right)
\]
unpow241.0%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot 0.125\right)\right)
\]
unpow241.0%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot 0.125\right)\right)
\]
times-frac56.8%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot 0.125\right)\right)
\]
*-commutative56.8%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\color{blue}{h \cdot {M}^{2}}}{\ell} \cdot 0.125\right)\right)
\]
unpow256.8%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot 0.125\right)\right)
\]
associate-*r*59.0%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell} \cdot 0.125\right)\right)
\]
associate-/l*62.8%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}} \cdot 0.125\right)\right)
\]
Simplified62.8%
\[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot 0.125\right)}\right)
\]
Step-by-step derivation pow162.8%
\[\leadsto \color{blue}{{\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot 0.125\right)\right)\right)}^{1}}
\]
pow-prod-down54.3%
\[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot 0.125\right)\right)\right)}^{1}
\]
metadata-eval54.3%
\[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot 0.125\right)\right)\right)}^{1}
\]
pow254.3%
\[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot 0.125\right)\right)\right)}^{1}
\]
associate-/l*52.9%
\[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{h}{\frac{\frac{\ell}{M}}{M}}} \cdot 0.125\right)\right)\right)}^{1}
\]
Applied egg-rr 52.9%
\[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\frac{\frac{\ell}{M}}{M}} \cdot 0.125\right)\right)\right)}^{1}}
\]
Step-by-step derivation unpow152.9%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\frac{\frac{\ell}{M}}{M}} \cdot 0.125\right)\right)}
\]
unpow1/252.9%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\frac{\frac{\ell}{M}}{M}} \cdot 0.125\right)\right)
\]
*-commutative52.9%
\[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(0.125 \cdot \frac{h}{\frac{\frac{\ell}{M}}{M}}\right)}\right)
\]
associate-/r/55.6%
\[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(0.125 \cdot \color{blue}{\left(\frac{h}{\frac{\ell}{M}} \cdot M\right)}\right)\right)
\]
Simplified55.6%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(0.125 \cdot \left(\frac{h}{\frac{\ell}{M}} \cdot M\right)\right)\right)}
\]
if 2.3499999999999998e-301 < l < 1.34999999999999999e100 Initial program 79.9%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u34.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef26.8%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 16.8%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def23.6%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p63.8%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg63.8%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/263.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative63.8%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in63.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval63.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/63.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified63.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around 0 81.0%
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation *-commutative81.0%
\[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative81.0%
\[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-181.0%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval81.0%
\[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr81.0%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square81.0%
\[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt81.0%
\[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr81.0%
\[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt81.0%
\[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified81.0%
\[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if 1.34999999999999999e100 < l Initial program 55.5%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 47.6%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u47.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/213.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 3 regimes into one program. Final simplification64.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq 2.35 \cdot 10^{-301}:\\
\;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d}\right)}^{2} \cdot \left(0.125 \cdot \left(M \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\\
\mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+100}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 12: 64.4% accurate, 1.5× speedup? \[\begin{array}{l}
t_0 := 1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\\
\mathbf{if}\;\ell \leq 1.6 \cdot 10^{-214}:\\
\;\;\;\;t_0 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\
\mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+99}:\\
\;\;\;\;t_0 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if l < 1.60000000000000007e-214 Initial program 73.1%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u32.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef27.1%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 23.0%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def26.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p62.8%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg62.8%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/262.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative62.8%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in62.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval62.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/62.8%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified62.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
if 1.60000000000000007e-214 < l < 8.40000000000000041e99 Initial program 78.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation expm1-log1p-u35.1%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}
\]
expm1-udef25.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1}
\]
Applied egg-rr 15.9%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)} - 1}
\]
Step-by-step derivation expm1-def24.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)\right)}
\]
expm1-log1p61.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)}
\]
sub-neg61.4%
\[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \color{blue}{\left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)}
\]
unpow1/261.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
*-commutative61.4%
\[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + \left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)\right)
\]
distribute-lft-neg-in61.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left(-0.5\right) \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)}\right)
\]
metadata-eval61.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\right)
\]
associate-/l/61.4%
\[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2}\right)\right)
\]
Simplified61.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)}
\]
Taylor expanded in d around 0 84.2%
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Step-by-step derivation *-commutative84.2%
\[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
*-commutative84.2%
\[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
unpow-184.2%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
metadata-eval84.2%
\[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
pow-sqr84.2%
\[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-sqrt-square84.2%
\[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt84.2%
\[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
fabs-sqr84.2%
\[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
rem-square-sqrt84.2%
\[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
Simplified84.2%
\[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)
\]
if 8.40000000000000041e99 < l Initial program 55.5%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 47.6%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u47.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/213.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval13.7%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 13.7%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified47.6%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 62.7%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 3 regimes into one program. Final simplification68.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq 1.6 \cdot 10^{-214}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\
\mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+99}:\\
\;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 13: 47.0% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;d \leq 9.5 \cdot 10^{-285}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;d \leq 2.4 \cdot 10^{+27}:\\
\;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\
\end{array}
\]
Derivation Split input into 3 regimes if d < 9.4999999999999997e-285 Initial program 68.8%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*68.8%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval68.8%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/268.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/268.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right)
\]
fma-def68.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right)
\]
Simplified68.7%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)}
\]
Taylor expanded in h around 0 34.7%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right)
\]
if 9.4999999999999997e-285 < d < 2.39999999999999998e27 Initial program 69.0%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 48.1%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative48.1%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*48.1%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*46.7%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/47.9%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow247.9%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*49.3%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow249.3%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified49.3%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 48.1%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow248.1%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/47.9%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow247.9%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/49.3%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*49.6%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative49.6%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified49.6%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Taylor expanded in D around 0 48.1%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative48.1%
\[\leadsto -0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}
\]
unpow248.1%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right)
\]
unpow248.1%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right)
\]
unswap-sqr55.8%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d}\right)
\]
*-commutative55.8%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot M\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d}\right)
\]
associate-*r/55.8%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{M \cdot D}{d}\right)}\right)
\]
associate-*r/53.0%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right)
\]
associate-*r*52.8%
\[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(D \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}\right)
\]
*-commutative52.8%
\[\leadsto -0.125 \cdot \color{blue}{\left(\left(D \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
associate-*l*54.2%
\[\leadsto -0.125 \cdot \color{blue}{\left(D \cdot \left(\left(M \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}
\]
associate-*r*51.0%
\[\leadsto -0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)
\]
unpow251.0%
\[\leadsto -0.125 \cdot \left(D \cdot \left(\left(\color{blue}{{M}^{2}} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)
\]
*-commutative51.0%
\[\leadsto -0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)
\]
associate-/r/49.6%
\[\leadsto -0.125 \cdot \left(D \cdot \left(\color{blue}{\frac{D}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)
\]
unpow249.6%
\[\leadsto -0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)
\]
Simplified49.6%
\[\leadsto \color{blue}{-0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}
\]
if 2.39999999999999998e27 < d Initial program 80.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 63.3%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation sqrt-div63.3%
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d
\]
metadata-eval63.3%
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d
\]
*-commutative63.3%
\[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 63.3%
\[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d
\]
Step-by-step derivation sqrt-prod75.4%
\[\leadsto \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot d
\]
Applied egg-rr 75.4%
\[\leadsto \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot d
\]
Recombined 3 regimes into one program. Final simplification47.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq 9.5 \cdot 10^{-285}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;d \leq 2.4 \cdot 10^{+27}:\\
\;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\
\end{array}
\]
Alternative 14: 49.9% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;d \leq 5.2 \cdot 10^{+30}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\
\end{array}
\]
Derivation Split input into 3 regimes if d < -1.999999999999994e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*71.4%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval71.4%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/271.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right)
\]
fma-def71.4%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right)
\]
Simplified71.4%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)}
\]
Taylor expanded in h around 0 35.2%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right)
\]
if -1.999999999999994e-310 < d < 5.19999999999999977e30 Initial program 64.1%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around 0 47.5%
\[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}
\]
Step-by-step derivation *-commutative47.5%
\[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}
\]
associate-*l*47.5%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
associate-/l*46.2%
\[\leadsto \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/r/46.0%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow246.0%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-/l*47.3%
\[\leadsto \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow247.3%
\[\leadsto \left(\frac{D}{\frac{d}{D}} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified47.3%
\[\leadsto \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}
\]
Taylor expanded in D around 0 47.5%
\[\leadsto \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation unpow247.5%
\[\leadsto \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l/46.0%
\[\leadsto \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \left(M \cdot M\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
unpow246.0%
\[\leadsto \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*r/47.3%
\[\leadsto \left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot \left(M \cdot M\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
associate-*l*47.5%
\[\leadsto \color{blue}{\left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
*-commutative47.5%
\[\leadsto \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Simplified47.5%
\[\leadsto \color{blue}{\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)
\]
Step-by-step derivation sqrt-div50.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Applied egg-rr 50.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot -0.125\right)
\]
Step-by-step derivation sqr-pow50.4%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot -0.125\right)
\]
rem-sqrt-square56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot -0.125\right)
\]
sqr-pow56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot -0.125\right)
\]
fabs-sqr56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot -0.125\right)
\]
sqr-pow56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot -0.125\right)
\]
metadata-eval56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right)
\]
Simplified56.0%
\[\leadsto \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right)
\]
if 5.19999999999999977e30 < d Initial program 80.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 63.3%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation sqrt-div63.3%
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d
\]
metadata-eval63.3%
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d
\]
*-commutative63.3%
\[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 63.3%
\[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d
\]
Step-by-step derivation sqrt-prod75.4%
\[\leadsto \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot d
\]
Applied egg-rr 75.4%
\[\leadsto \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot d
\]
Recombined 3 regimes into one program. Final simplification49.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{elif}\;d \leq 5.2 \cdot 10^{+30}:\\
\;\;\;\;\left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\
\mathbf{else}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\
\end{array}
\]
Alternative 15: 32.1% accurate, 1.6× speedup? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -4.999999999999985e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 12.5%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation add-cbrt-cube14.7%
\[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d
\]
pow1/314.7%
\[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333}} \cdot d
\]
add-sqr-sqrt14.7%
\[\leadsto {\left(\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333} \cdot d
\]
pow114.7%
\[\leadsto {\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)}^{0.3333333333333333} \cdot d
\]
pow1/214.7%
\[\leadsto {\left({\left(\frac{1}{\ell \cdot h}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)}^{0.3333333333333333} \cdot d
\]
pow-prod-up14.7%
\[\leadsto {\color{blue}{\left({\left(\frac{1}{\ell \cdot h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \cdot d
\]
*-commutative14.7%
\[\leadsto {\left({\left(\frac{1}{\color{blue}{h \cdot \ell}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \cdot d
\]
metadata-eval14.7%
\[\leadsto {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot d
\]
Applied egg-rr 14.7%
\[\leadsto \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot d
\]
Step-by-step derivation unpow1/314.7%
\[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \cdot d
\]
Simplified14.7%
\[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \cdot d
\]
if -4.999999999999985e-310 < l Initial program 71.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 40.3%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/220.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 2 regimes into one program. Final simplification31.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 16: 31.9% accurate, 1.6× speedup? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -4.999999999999985e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 12.5%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation sqrt-div12.5%
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d
\]
metadata-eval12.5%
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d
\]
*-commutative12.5%
\[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 12.5%
\[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d
\]
Step-by-step derivation add-cbrt-cube14.7%
\[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \cdot d
\]
add-sqr-sqrt14.7%
\[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 14.7%
\[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(h \cdot \ell\right) \cdot \sqrt{h \cdot \ell}}}} \cdot d
\]
Step-by-step derivation *-commutative14.7%
\[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\sqrt{h \cdot \ell} \cdot \left(h \cdot \ell\right)}}} \cdot d
\]
unpow1/214.7%
\[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(h \cdot \ell\right)}} \cdot d
\]
pow-plus14.7%
\[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{\left(0.5 + 1\right)}}}} \cdot d
\]
metadata-eval14.7%
\[\leadsto \frac{1}{\sqrt[3]{{\left(h \cdot \ell\right)}^{\color{blue}{1.5}}}} \cdot d
\]
Simplified14.7%
\[\leadsto \frac{1}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \cdot d
\]
if -4.999999999999985e-310 < l Initial program 71.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 40.3%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/220.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 2 regimes into one program. Final simplification31.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;d \cdot \frac{1}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 17: 30.1% accurate, 1.6× speedup? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -4.999999999999985e-310 Initial program 71.4%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 12.5%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation sqrt-div12.5%
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d
\]
metadata-eval12.5%
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d
\]
*-commutative12.5%
\[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 12.5%
\[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d
\]
Step-by-step derivation associate-*l/12.5%
\[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}}
\]
*-un-lft-identity12.5%
\[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}
\]
Applied egg-rr 12.5%
\[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}
\]
if -4.999999999999985e-310 < l Initial program 71.2%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 40.3%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/220.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval20.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 20.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def39.4%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified40.3%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 47.8%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 2 regimes into one program. Final simplification29.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\
\;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 18: 45.8% accurate, 1.6× speedup? \[\begin{array}{l}
\mathbf{if}\;d \leq 1.6 \cdot 10^{-246}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if d < 1.6e-246 Initial program 67.7%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Step-by-step derivation associate-*l*67.8%
\[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}
\]
metadata-eval67.8%
\[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/267.8%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
metadata-eval67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
unpow1/267.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)
\]
sub-neg67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)
\]
+-commutative67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right)
\]
*-commutative67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right)
\]
distribute-rgt-neg-in67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right)
\]
fma-def67.8%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right)
\]
Simplified67.7%
\[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)}
\]
Taylor expanded in h around 0 33.5%
\[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right)
\]
if 1.6e-246 < d Initial program 75.7%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 42.8%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u41.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef21.0%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/221.0%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow21.0%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow21.0%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative21.0%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval21.0%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 21.0%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def41.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p42.8%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified42.8%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Step-by-step derivation unpow-prod-down50.9%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Applied egg-rr 50.9%
\[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d
\]
Recombined 2 regimes into one program. Final simplification41.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \leq 1.6 \cdot 10^{-246}:\\
\;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\
\end{array}
\]
Alternative 19: 26.4% accurate, 3.1× speedup? \[d \cdot \sqrt{\frac{1}{h \cdot \ell}}
\]
Derivation Initial program 71.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 26.2%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Final simplification26.2%
\[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}}
\]
Alternative 20: 26.2% accurate, 3.1× speedup? \[d \cdot {\left(h \cdot \ell\right)}^{-0.5}
\]
Derivation Initial program 71.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 26.2%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation expm1-log1p-u25.8%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d
\]
expm1-udef16.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d
\]
pow1/216.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}\right)} - 1\right) \cdot d
\]
inv-pow16.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot d
\]
pow-pow16.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot d
\]
*-commutative16.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)} - 1\right) \cdot d
\]
metadata-eval16.3%
\[\leadsto \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d
\]
Applied egg-rr 16.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \cdot d
\]
Step-by-step derivation expm1-def25.8%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot d
\]
expm1-log1p26.2%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Simplified26.2%
\[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d
\]
Final simplification26.2%
\[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5}
\]
Alternative 21: 26.2% accurate, 3.2× speedup? \[\frac{d}{\sqrt{h \cdot \ell}}
\]
Derivation Initial program 71.3%
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
\]
Taylor expanded in d around inf 26.2%
\[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}
\]
Step-by-step derivation sqrt-div26.1%
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d
\]
metadata-eval26.1%
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d
\]
*-commutative26.1%
\[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d
\]
Applied egg-rr 26.1%
\[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d
\]
Step-by-step derivation associate-*l/26.1%
\[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}}
\]
*-un-lft-identity26.1%
\[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}
\]
Applied egg-rr 26.1%
\[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}
\]
Final simplification26.1%
\[\leadsto \frac{d}{\sqrt{h \cdot \ell}}
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (d h l M D)
:name "Henrywood and Agarwal, Equation (12)"
:precision binary64
(* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
