Original	25.8
Target	0.4
Herbie	8.0

Derivation

Split input into 2 regimes
if d < 3.7092653838509206e+139
1. Initial program 23.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification23.1
  \[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt23.2
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity23.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac23.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified23.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified14.8
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/14.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified14.7
  \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Using strategy rm
13. Applied add-sqr-sqrt14.9
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
14. Applied associate-/r*14.9
  \[\leadsto \color{blue}{\frac{\frac{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
15. Using strategy rm
16. Applied div-sub14.9
  \[\leadsto \frac{\frac{\color{blue}{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*} - \frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}\]
17. Applied div-sub14.9
  \[\leadsto \frac{\color{blue}{\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}} - \frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}}{\sqrt{\sqrt{d^2 + c^2}^*}}\]
18. Applied div-sub14.9
  \[\leadsto \color{blue}{\frac{\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}} - \frac{\frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
19. Simplified7.0
  \[\leadsto \color{blue}{\frac{\frac{b}{\sqrt{d^2 + c^2}^*}}{\frac{\sqrt{d^2 + c^2}^*}{c}}} - \frac{\frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}\]
if 3.7092653838509206e+139 < d
1. Initial program 42.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification42.1
  \[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
3. Using strategy rm
4. Applied add-sqr-sqrt42.1
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity42.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac42.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Simplified42.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Simplified27.3
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*l/27.2
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
11. Simplified27.2
  \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around 0 14.3
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified14.3
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 2 regimes into one program.
Final simplification8.0
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le 3.7092653838509206 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{d^2 + c^2}^*}}{\frac{\sqrt{d^2 + c^2}^*}{c}} - \frac{\frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

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Target

Derivation

`if d < 3.7092653838509206e+139`

`if 3.7092653838509206e+139 < d`

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