Original	26.3
Target	0.5
Herbie	13.0

Derivation

Split input into 3 regimes
if c < -1.2534913427767432e+147
1. Initial program 44.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt44.1
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity44.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac44.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified44.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified27.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/27.9
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified27.9
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied clear-num27.9
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{(d \cdot b + \left(c \cdot a\right))_*}}}}{\sqrt{c^2 + d^2}^*}\]
13. Taylor expanded around -inf 13.1
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c^2 + d^2}^*}\]
14. Simplified13.1
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{c^2 + d^2}^*}\]
if -1.2534913427767432e+147 < c < 2.135523821820183e+102
1. Initial program 19.0
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt19.0
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity19.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac19.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified19.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified11.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/11.8
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified11.8
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied clear-num11.9
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{(d \cdot b + \left(c \cdot a\right))_*}}}}{\sqrt{c^2 + d^2}^*}\]
if 2.135523821820183e+102 < c
1. Initial program 39.8
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt39.8
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity39.8
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac39.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified39.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified26.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/26.0
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified26.0
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Taylor expanded around 0 16.6
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{c^2 + d^2}^*}\]
Recombined 3 regimes into one program.
Final simplification13.0
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.2534913427767432 \cdot 10^{+147}:\\ \;\;\;\;\frac{-a}{\sqrt{c^2 + d^2}^*}\\ \mathbf{elif}\;c \le 2.135523821820183 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{(d \cdot b + \left(a \cdot c\right))_*}}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -1.2534913427767432e+147`

`if -1.2534913427767432e+147 < c < 2.135523821820183e+102`

`if 2.135523821820183e+102 < c`

Runtime