Original	25.7
Target	0.4
Herbie	12.8

Derivation

Split input into 3 regimes
if c < -4.0242236830456733e+121
1. Initial program 41.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt41.4
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity41.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac41.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify41.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify28.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*r/28.0
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
10. Applied simplify28.0
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Taylor expanded around -inf 15.3
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c^2 + d^2}^*}\]
12. Applied simplify15.3
  \[\leadsto \color{blue}{\frac{-a}{\sqrt{c^2 + d^2}^*}}\]
if -4.0242236830456733e+121 < c < 1.6578044070822745e+141
1. Initial program 18.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.4
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity18.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify12.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if 1.6578044070822745e+141 < c
1. Initial program 43.9
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt43.9
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity43.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac43.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify43.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify29.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around 0 14.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{a}\]
9. Applied simplify14.0
  \[\leadsto \color{blue}{\frac{a}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -4.0242236830456733e+121`

`if -4.0242236830456733e+121 < c < 1.6578044070822745e+141`

`if 1.6578044070822745e+141 < c`

Runtime