Original	25.9
Target	0.4
Herbie	13.1

Derivation

Split input into 3 regimes
if d < -1.376753448750176e+191
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 simplify30.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around 0 11.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{a}\]
9. Applied simplify11.5
  \[\leadsto \color{blue}{\frac{a}{\sqrt{c^2 + d^2}^*}}\]
if -1.376753448750176e+191 < d < 0.1841955643280176
1. Initial program 20.4
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt20.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity20.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac20.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify20.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
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 clear-num12.5
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{c \cdot b - d \cdot a}}}\]
if 0.1841955643280176 < d
1. Initial program 33.0
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt33.0
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity33.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac33.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify33.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify22.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*r/22.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(c \cdot b - d \cdot a\right)}{\sqrt{c^2 + d^2}^*}}\]
10. Applied simplify22.1
  \[\leadsto \frac{\color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Taylor expanded around 0 29.9
  \[\leadsto \frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\color{blue}{d}}\]
12. Applied simplify15.1
  \[\leadsto \color{blue}{\frac{(\left(\frac{c}{d}\right) \cdot b + \left(-a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if d < -1.376753448750176e+191`

`if -1.376753448750176e+191 < d < 0.1841955643280176`

`if 0.1841955643280176 < d`

Runtime