Original	25.4
Target	0.5
Herbie	10.9

Derivation

Split input into 3 regimes
if c < -1.182260775010774e+97
1. Initial program 38.7
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify38.7
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt38.7
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity38.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac38.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify38.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify25.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied add-sqr-sqrt25.1
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
11. Applied associate-/r*25.1
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
12. Taylor expanded around -inf 14.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(\frac{d \cdot a}{c} - b\right)}\]
13. Applied simplify10.2
  \[\leadsto \color{blue}{\frac{(a \cdot \left(\frac{d}{c}\right) + \left(-b\right))_*}{\sqrt{d^2 + c^2}^*}}\]
if -1.182260775010774e+97 < c < 8.927282505690149e+75
1. Initial program 17.7
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify17.7
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt17.7
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity17.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac17.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify17.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify11.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*r/11.4
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
11. Applied simplify11.3
  \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 8.927282505690149e+75 < c
1. Initial program 37.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify37.1
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt37.1
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity37.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac37.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify37.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify24.9
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied add-sqr-sqrt25.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
11. Applied associate-/r*25.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
12. Taylor expanded around inf 13.9
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(b - \frac{d \cdot a}{c}\right)}\]
13. Applied simplify10.4
  \[\leadsto \color{blue}{\frac{(\left(\frac{d}{c}\right) \cdot \left(-a\right) + b)_*}{\sqrt{d^2 + c^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -1.182260775010774e+97`

`if -1.182260775010774e+97 < c < 8.927282505690149e+75`

`if 8.927282505690149e+75 < c`

Runtime