Original	25.7
Target	0.5
Herbie	11.4

Derivation

Split input into 3 regimes
if c < -1.4085172178463586e+108
1. Initial program 40.9
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt40.9
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity40.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac40.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify40.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify27.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt28.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied *-un-lft-identity28.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{\color{blue}{1 \cdot (b \cdot d + \left(c \cdot a\right))_*}}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}\]
11. Applied times-frac28.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}\right)}\]
12. Taylor expanded around -inf 12.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-\left(\frac{b \cdot d}{c} + a\right)\right)}\]
13. Applied simplify8.7
  \[\leadsto \color{blue}{\frac{(\left(\frac{b}{c}\right) \cdot \left(-d\right) + \left(-a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if -1.4085172178463586e+108 < c < 3.2092839043956334e-17
1. Initial program 17.9
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt17.9
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity17.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac17.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify17.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify10.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if 3.2092839043956334e-17 < c
1. Initial program 31.6
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt31.6
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity31.6
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac31.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify31.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify21.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt21.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied *-un-lft-identity21.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{\color{blue}{1 \cdot (b \cdot d + \left(c \cdot a\right))_*}}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}\]
11. Applied times-frac21.8
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}\right)}\]
12. Taylor expanded around inf 17.4
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{b \cdot d}{c} + a\right)}\]
13. Applied simplify15.2
  \[\leadsto \color{blue}{\frac{(d \cdot \left(\frac{b}{c}\right) + a)_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -1.4085172178463586e+108`

`if -1.4085172178463586e+108 < c < 3.2092839043956334e-17`

`if 3.2092839043956334e-17 < c`

Runtime