Original	25.0
Target	0.4
Herbie	12.8

Derivation

Split input into 3 regimes
if c < -7.349917155713805e+149
1. Initial program 42.8
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt42.8
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity42.8
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac42.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified42.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified27.5
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around -inf 14.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-1 \cdot a\right)}\]
9. Simplified14.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-a\right)}\]
if -7.349917155713805e+149 < c < 2.6296693594638745e+199
1. Initial program 20.0
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt20.0
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity20.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac20.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified20.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified13.1
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/13.0
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified13.0
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
if 2.6296693594638745e+199 < c
1. Initial program 40.4
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt40.4
  \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity40.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac40.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Simplified40.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified26.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/26.7
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
10. Simplified26.6
  \[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
11. Using strategy rm
12. Applied add-sqr-sqrt26.7
  \[\leadsto \frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
13. Applied associate-/r*26.7
  \[\leadsto \frac{\color{blue}{\frac{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{\sqrt{c^2 + d^2}^*}}}{\sqrt{\sqrt{c^2 + d^2}^*}}}}{\sqrt{c^2 + d^2}^*}\]
14. Taylor expanded around 0 9.4
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{c^2 + d^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.8
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -7.349917155713805 \cdot 10^{+149}:\\ \;\;\;\;\frac{-1}{\sqrt{c^2 + d^2}^*} \cdot a\\ \mathbf{elif}\;c \le 2.6296693594638745 \cdot 10^{+199}:\\ \;\;\;\;\frac{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -7.349917155713805e+149`

`if -7.349917155713805e+149 < c < 2.6296693594638745e+199`

`if 2.6296693594638745e+199 < c`

Runtime