Original	25.4
Target	0.4
Herbie	10.4

Derivation

Split input into 3 regimes
if c < -6.145662328138862e+103
1. Initial program 38.7
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt38.7
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity38.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac38.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify38.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify25.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt25.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied associate-/r*25.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{\frac{c \cdot b - a \cdot d}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
11. Taylor expanded around -inf 12.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{d \cdot a}{c} - b\right)}\]
12. Applied simplify9.1
  \[\leadsto \color{blue}{\frac{(a \cdot \left(\frac{d}{c}\right) + \left(-b\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if -6.145662328138862e+103 < c < 7.19883100474498e+116
1. Initial program 18.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.1
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity18.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac18.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify18.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify11.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
if 7.19883100474498e+116 < c
1. Initial program 40.4
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt40.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity40.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac40.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify40.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify25.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt25.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied associate-/r*25.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{\frac{c \cdot b - a \cdot d}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
11. Taylor expanded around inf 12.5
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(b - \frac{d \cdot a}{c}\right)}\]
12. Applied simplify8.8
  \[\leadsto \color{blue}{\frac{(\left(\frac{d}{c}\right) \cdot \left(-a\right) + b)_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -6.145662328138862e+103`

`if -6.145662328138862e+103 < c < 7.19883100474498e+116`

`if 7.19883100474498e+116 < c`

Runtime