Original	25.8
Target	0.5
Herbie	13.1

Derivation

Split input into 3 regimes
if c < -1.0108561403615378e+75
1. Initial program 36.9
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt36.9
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity36.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac36.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified36.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified25.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/25.6
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified25.6
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied *-un-lft-identity25.6
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (d \cdot b + \left(c \cdot a\right))_*}}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
13. Applied associate-/l*25.6
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{(d \cdot b + \left(c \cdot a\right))_*}}}}{\sqrt{c^2 + d^2}^*}\]
14. Taylor expanded around -inf 16.3
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c^2 + d^2}^*}\]
15. Simplified16.3
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{c^2 + d^2}^*}\]
if -1.0108561403615378e+75 < c < 7.020069000953778e+131
1. Initial program 18.5
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.5
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity18.5
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac18.5
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified18.5
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified11.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/11.3
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified11.3
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied *-un-lft-identity11.3
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (d \cdot b + \left(c \cdot a\right))_*}}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
13. Applied associate-/l*11.4
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{(d \cdot b + \left(c \cdot a\right))_*}}}}{\sqrt{c^2 + d^2}^*}\]
14. Using strategy rm
15. Applied div-inv11.6
  \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt{c^2 + d^2}^* \cdot \frac{1}{(d \cdot b + \left(c \cdot a\right))_*}}}}{\sqrt{c^2 + d^2}^*}\]
if 7.020069000953778e+131 < c
1. Initial program 42.7
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt42.7
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity42.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac42.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified42.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified28.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/28.5
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified28.5
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Taylor expanded around 0 15.4
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{c^2 + d^2}^*}\]
Recombined 3 regimes into one program.
Final simplification13.1
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.0108561403615378 \cdot 10^{+75}:\\ \;\;\;\;\frac{-a}{\sqrt{c^2 + d^2}^*}\\ \mathbf{elif}\;c \le 7.020069000953778 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{(d \cdot b + \left(a \cdot c\right))_*} \cdot \sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -1.0108561403615378e+75`

`if -1.0108561403615378e+75 < c < 7.020069000953778e+131`

`if 7.020069000953778e+131 < c`

Runtime