Original	25.4
Target	0.4
Herbie	12.6

Derivation

Split input into 3 regimes
if d < -3.4987512270458407e+136
1. Initial program 41.0
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification41.0
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt41.0
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity41.0
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac41.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified41.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified26.6
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/26.5
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified26.5
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity26.5
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*26.6
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
15. Taylor expanded around -inf 14.1
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
16. Simplified14.1
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]
if -3.4987512270458407e+136 < d < 7.972923326799236e+198
1. Initial program 20.7
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification20.7
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt20.7
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity20.7
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac20.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified20.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified12.8
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/12.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified12.7
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 7.972923326799236e+198 < d
1. Initial program 39.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification39.4
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt39.4
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity39.4
  \[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac39.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified39.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified28.8
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/28.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified28.7
  \[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied *-un-lft-identity28.7
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot (c \cdot a + \left(b \cdot d\right))_*}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
14. Applied associate-/l*28.7
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(c \cdot a + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]
15. Taylor expanded around inf 9.5
  \[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.6
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le -3.4987512270458407 \cdot 10^{+136}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;d \le 7.972923326799236 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if d < -3.4987512270458407e+136`

`if -3.4987512270458407e+136 < d < 7.972923326799236e+198`

`if 7.972923326799236e+198 < d`

Runtime