Original	25.9
Target	0.5
Herbie	11.2

Derivation

Split input into 3 regimes
if c < -2.227620296543906e+84
1. Initial program 37.2
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt37.2
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity37.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac37.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify37.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify25.0
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt25.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied *-un-lft-identity25.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{\color{blue}{1 \cdot (b \cdot d + \left(c \cdot a\right))_*}}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}\]
11. Applied times-frac25.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}\right)}\]
12. Taylor expanded around -inf 14.5
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-\left(\frac{b \cdot d}{c} + a\right)\right)}\]
13. Applied simplify10.9
  \[\leadsto \color{blue}{\frac{(\left(\frac{b}{c}\right) \cdot \left(-d\right) + \left(-a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if -2.227620296543906e+84 < c < 2.168394667707026e+41
1. Initial program 18.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.3
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity18.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify11.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if 2.168394667707026e+41 < c
1. Initial program 35.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt35.4
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity35.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac35.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify35.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify23.9
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt24.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}}\]
10. Applied *-un-lft-identity24.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{\color{blue}{1 \cdot (b \cdot d + \left(c \cdot a\right))_*}}{\left(\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}\right) \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}}\]
11. Applied times-frac24.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{c^2 + d^2}^*} \cdot \sqrt[3]{\sqrt{c^2 + d^2}^*}} \cdot \frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt[3]{\sqrt{c^2 + d^2}^*}}\right)}\]
12. Taylor expanded around inf 15.2
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{b \cdot d}{c} + a\right)}\]
13. Applied simplify11.9
  \[\leadsto \color{blue}{\frac{(d \cdot \left(\frac{b}{c}\right) + a)_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -2.227620296543906e+84`

`if -2.227620296543906e+84 < c < 2.168394667707026e+41`

`if 2.168394667707026e+41 < c`

Runtime