Original	25.9
Target	0.4
Herbie	11.7

Derivation

Split input into 3 regimes
if c < -4.533196055403128e+161
1. Initial program 44.3
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt44.3
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity44.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac44.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify44.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify28.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around -inf 12.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-1 \cdot b\right)}\]
9. Applied simplify12.7
  \[\leadsto \color{blue}{\frac{-b}{\sqrt{c^2 + d^2}^*}}\]
if -4.533196055403128e+161 < c < 4.170741103824391e+136
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 simplify12.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*r/12.4
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(c \cdot b - d \cdot a\right)}{\sqrt{c^2 + d^2}^*}}\]
10. Applied simplify12.3
  \[\leadsto \frac{\color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
if 4.170741103824391e+136 < c
1. Initial program 42.6
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt42.6
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity42.6
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac42.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify42.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify28.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around inf 30.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - d \cdot a}{\color{blue}{c}}\]
9. Applied simplify7.5
  \[\leadsto \color{blue}{\frac{(\left(\frac{a}{c}\right) \cdot \left(-d\right) + b)_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -4.533196055403128e+161`

`if -4.533196055403128e+161 < c < 4.170741103824391e+136`

`if 4.170741103824391e+136 < c`

Runtime