Original	26.1
Target	0.5
Herbie	13.3

Derivation

Split input into 3 regimes
if c < -4.086380967133727e+141
1. Initial program 43.2
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification43.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.2
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified43.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified28.7
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/28.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified28.6
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 14.5
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified14.5
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -4.086380967133727e+141 < c < 1.5389869704006215e+74
1. Initial program 19.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification19.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt19.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity19.1
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac19.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified19.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified11.7
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/11.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified11.6
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied add-sqr-sqrt11.8
  \[\leadsto \frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
14. Applied add-sqr-sqrt12.0
  \[\leadsto \frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}\]
15. Applied *-un-lft-identity12.0
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}\]
16. Applied times-frac12.1
  \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{\sqrt{d^2 + c^2}^*}}}}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}\]
17. Applied times-frac12.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
18. Simplified11.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}\]
19. Simplified11.7
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{d^2 + c^2}^*}}\]
20. Using strategy rm
21. Applied associate-*r/11.7
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot (d \cdot b + \left(c \cdot a\right))_*}{\sqrt{d^2 + c^2}^*}}\]
if 1.5389869704006215e+74 < c
1. Initial program 36.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification36.4
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt36.4
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity36.4
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac36.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified36.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified24.8
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/24.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified24.7
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around inf 17.9
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification13.3
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -4.086380967133727 \cdot 10^{+141}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.5389869704006215 \cdot 10^{+74}:\\ \;\;\;\;\frac{(d \cdot b + \left(a \cdot c\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -4.086380967133727e+141`

`if -4.086380967133727e+141 < c < 1.5389869704006215e+74`

`if 1.5389869704006215e+74 < c`

Runtime