Original	25.9
Target	0.5
Herbie	12.7

Derivation

Split input into 3 regimes
if c < -1.953014043856611e+167
1. Initial program 43.7
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt43.7
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity43.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac43.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify43.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify29.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around -inf 11.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-1 \cdot b\right)}\]
9. Applied simplify11.2
  \[\leadsto \color{blue}{\frac{-b}{\sqrt{c^2 + d^2}^*}}\]
if -1.953014043856611e+167 < c < 2.6887306195946796e+124
1. Initial program 19.3
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt19.3
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity19.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac19.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify19.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify12.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-sqr-sqrt12.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}}\]
10. Applied *-un-lft-identity12.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}\]
11. Applied times-frac12.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{c^2 + d^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{\sqrt{c^2 + d^2}^*}}\right)}\]
if 2.6887306195946796e+124 < c
1. Initial program 42.2
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt42.2
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity42.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac42.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify42.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify27.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around inf 15.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{b}\]
9. Applied simplify15.3
  \[\leadsto \color{blue}{\frac{b}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -1.953014043856611e+167`

`if -1.953014043856611e+167 < c < 2.6887306195946796e+124`

`if 2.6887306195946796e+124 < c`

Runtime