Original	25.7
Target	0.5
Herbie	13.2

Derivation

Split input into 3 regimes
if c < -1.4416367100968121e+88
1. Initial program 37.8
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification37.8
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt37.8
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity37.8
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac37.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified37.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified24.6
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/24.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified24.5
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 17.1
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified17.1
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -1.4416367100968121e+88 < c < 5.889595329217942e+199
1. Initial program 20.2
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification20.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt20.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity20.2
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac20.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified20.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.2
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*r/12.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot (d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
11. Using strategy rm
12. Applied *-un-lft-identity12.1
  \[\leadsto \frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot (d \cdot b + \left(a \cdot c\right))_*}{\color{blue}{1 \cdot \sqrt{d^2 + c^2}^*}}\]
13. Applied times-frac12.2
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*}}{1} \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
14. Simplified12.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}\]
if 5.889595329217942e+199 < c
1. Initial program 43.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification43.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.1
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.1
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified43.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified32.2
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/32.2
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified32.2
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around inf 12.6
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification13.2
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.4416367100968121 \cdot 10^{+88}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 5.889595329217942 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -1.4416367100968121e+88`

`if -1.4416367100968121e+88 < c < 5.889595329217942e+199`

`if 5.889595329217942e+199 < c`

Runtime