Original	26.4
Target	0.4
Herbie	26.4

Derivation

Split input into 2 regimes
if c < -1.7386260483318646e+62
1. Initial program 37.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification37.3
  \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt37.3
  \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*37.2
  \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Using strategy rm
7. Applied add-sqr-sqrt37.3
  \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\color{blue}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}\]
8. Applied add-sqr-sqrt37.3
  \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]
9. Applied sqrt-prod37.4
  \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]
10. Applied *-un-lft-identity37.4
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(b \cdot d + a \cdot c\right)}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]
11. Applied times-frac37.4
  \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}\]
12. Applied times-frac37.4
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}\]
13. Simplified37.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}}} \cdot \frac{\frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}\]
14. Simplified37.3
  \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
15. Taylor expanded around -inf 37.6
  \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\left(-1 \cdot a\right)}\]
16. Simplified37.6
  \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\left(-a\right)}\]
if -1.7386260483318646e+62 < c
1. Initial program 23.5
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification23.5
  \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt23.5
  \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*23.4
  \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
Recombined 2 regimes into one program.
Final simplification26.4
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.7386260483318646 \cdot 10^{+62}:\\ \;\;\;\;a \cdot \frac{-1}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot b + c \cdot a}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Error

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Target

Derivation

`if c < -1.7386260483318646e+62`

`if -1.7386260483318646e+62 < c`

Runtime

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