Original	26.5
Target	0.4
Herbie	13.6

Derivation

Split input into 3 regimes
if c < -1.1311343419872614e+58
1. Initial program 36.2
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify36.2
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt36.2
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*36.2
  \[\leadsto \color{blue}{\frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef36.2
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def36.2
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Taylor expanded around -inf 18.3
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
10. Applied simplify18.3
  \[\leadsto \color{blue}{\frac{-b}{\sqrt{d^2 + c^2}^*}}\]
if -1.1311343419872614e+58 < c < 4.330565057877467e+130
1. Initial program 19.1
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify19.1
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt19.1
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*19.0
  \[\leadsto \color{blue}{\frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef19.0
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def19.0
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied fma-udef19.0
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{d^2 + c^2}^*}\]
11. Applied hypot-def11.6
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 4.330565057877467e+130 < c
1. Initial program 42.3
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Applied simplify42.3
  \[\leadsto \color{blue}{\frac{c \cdot b - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt42.3
  \[\leadsto \frac{c \cdot b - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*42.3
  \[\leadsto \color{blue}{\frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef42.3
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def42.3
  \[\leadsto \frac{\frac{c \cdot b - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Taylor expanded around inf 15.1
  \[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.

Error

Target

Derivation

`if c < -1.1311343419872614e+58`

`if -1.1311343419872614e+58 < c < 4.330565057877467e+130`

`if 4.330565057877467e+130 < c`

Runtime