Original	25.7
Target	0.5
Herbie	10.4

Derivation

Split input into 3 regimes
if c < -2.1803526342689262e+101
1. Initial program 39.8
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt39.8
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity39.8
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac39.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify39.8
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify27.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Taylor expanded around -inf 30.6
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\color{blue}{-1 \cdot c}}\]
9. Applied simplify9.9
  \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{-\sqrt{c^2 + d^2}^*}}\]
if -2.1803526342689262e+101 < c < 3.186542051972063e+128
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-frac18.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied 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 simplify11.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied pow111.3
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{{\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
10. Applied pow111.3
  \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*}\right)}^{1}} \cdot {\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}\]
11. Applied pow-prod-down11.3
  \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
12. Applied simplify11.2
  \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right)}}^{1}\]
if 3.186542051972063e+128 < c
1. Initial program 41.0
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Using strategy rm
3. Applied add-sqr-sqrt41.0
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
4. Applied *-un-lft-identity41.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
5. Applied times-frac41.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Applied simplify41.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
7. Applied simplify26.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
8. Using strategy rm
9. Applied pow126.7
  \[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{{\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
10. Applied pow126.7
  \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*}\right)}^{1}} \cdot {\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}\]
11. Applied pow-prod-down26.7
  \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
12. Applied simplify26.7
  \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right)}}^{1}\]
13. Taylor expanded around inf 29.2
  \[\leadsto {\left(\frac{\frac{b \cdot c - a \cdot d}{\color{blue}{c}}}{\sqrt{c^2 + d^2}^*}\right)}^{1}\]
14. Applied simplify8.0
  \[\leadsto \color{blue}{\frac{(\left(\frac{d}{c}\right) \cdot \left(-a\right) + b)_*}{\sqrt{c^2 + d^2}^*}}\]
Recombined 3 regimes into one program.
Applied simplify10.4
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;c \le -2.1803526342689262 \cdot 10^{+101}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{-\sqrt{c^2 + d^2}^*}\\ \mathbf{if}\;c \le 3.186542051972063 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{d}{c}\right) \cdot \left(-a\right) + b)_*}{\sqrt{c^2 + d^2}^*}\\ \end{array}}\]

Error

Target

Derivation

`if c < -2.1803526342689262e+101`

`if -2.1803526342689262e+101 < c < 3.186542051972063e+128`

`if 3.186542051972063e+128 < c`

Runtime