Original	25.6
Target	0.4
Herbie	12.5

Derivation

Split input into 3 regimes
if d < -5.6914378486354656e+169
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. Simplified27.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/27.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified27.7
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied add-sqr-sqrt27.8
  \[\leadsto \frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
13. Applied associate-/r*27.8
  \[\leadsto \frac{\color{blue}{\frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{\sqrt{c^2 + d^2}^*}}}{\sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
14. Taylor expanded around -inf 12.6
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c^2 + d^2}^*}\]
15. Simplified12.6
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{c^2 + d^2}^*}\]
if -5.6914378486354656e+169 < d < 1.4868474142989653e+141
1. Initial program 19.7
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt19.7
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity19.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac19.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified19.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified12.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/12.2
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified12.2
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
if 1.4868474142989653e+141 < d
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. Simplified26.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/26.8
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified26.8
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied add-sqr-sqrt26.9
  \[\leadsto \frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
13. Applied associate-/r*26.9
  \[\leadsto \frac{\color{blue}{\frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{\sqrt{c^2 + d^2}^*}}}{\sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
14. Taylor expanded around inf 13.7
  \[\leadsto \frac{\color{blue}{b}}{\sqrt{c^2 + d^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.5
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le -5.6914378486354656 \cdot 10^{+169}:\\ \;\;\;\;\frac{-b}{\sqrt{c^2 + d^2}^*}\\ \mathbf{elif}\;d \le 1.4868474142989653 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if d < -5.6914378486354656e+169`

`if -5.6914378486354656e+169 < d < 1.4868474142989653e+141`

`if 1.4868474142989653e+141 < d`

Runtime