Original	25.8
Target	0.5
Herbie	11.0

Derivation

Split input into 3 regimes
if c < -2.9830793252751902e+141
1. Initial program 43.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify43.6
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.6
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity43.6
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac43.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify43.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify28.0
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied add-sqr-sqrt28.1
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
11. Applied associate-/r*28.1
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
12. Taylor expanded around -inf 11.5
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(\frac{d \cdot a}{c} - b\right)}\]
13. Applied simplify8.1
  \[\leadsto \color{blue}{\frac{(a \cdot \left(\frac{d}{c}\right) + \left(-b\right))_*}{\sqrt{d^2 + c^2}^*}}\]
if -2.9830793252751902e+141 < c < 7.202117007000063e+42
1. Initial program 18.2
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify18.2
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt18.2
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity18.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac18.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify18.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify11.3
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied associate-*r/11.3
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
11. Applied simplify11.2
  \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 7.202117007000063e+42 < c
1. Initial program 36.2
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify36.2
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt36.2
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied *-un-lft-identity36.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
6. Applied times-frac36.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
7. Applied simplify36.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied simplify24.3
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied add-sqr-sqrt24.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
11. Applied associate-/r*24.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{\sqrt{d^2 + c^2}^*}}}\]
12. Taylor expanded around inf 15.4
  \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(b - \frac{d \cdot a}{c}\right)}\]
13. Applied simplify12.4
  \[\leadsto \color{blue}{\frac{(\left(\frac{d}{c}\right) \cdot \left(-a\right) + b)_*}{\sqrt{d^2 + c^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -2.9830793252751902e+141`

`if -2.9830793252751902e+141 < c < 7.202117007000063e+42`

`if 7.202117007000063e+42 < c`

Runtime