Original	25.6
Target	0.5
Herbie	12.7

Derivation

Split input into 3 regimes
if c < -2.5199309894899704e+102
1. Initial program 38.7
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified38.7
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt38.7
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*38.6
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef38.6
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied hypot-def38.6
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
9. Using strategy rm
10. Applied fma-udef38.6
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
11. Applied hypot-def25.1
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
12. Taylor expanded around -inf 16.5
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified16.5
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -2.5199309894899704e+102 < c < 1.8011763110697258e+137
1. Initial program 18.5
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified18.5
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt18.5
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*18.4
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef18.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied hypot-def18.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
9. Using strategy rm
10. Applied fma-udef18.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
11. Applied hypot-def11.5
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
if 1.8011763110697258e+137 < c
1. Initial program 42.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified42.3
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt42.3
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*42.3
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef42.3
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
8. Applied hypot-def42.3
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
9. Using strategy rm
10. Applied fma-udef42.3
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
11. Applied hypot-def27.5
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
12. Taylor expanded around inf 13.7
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.7
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.5199309894899704 \cdot 10^{+102}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.8011763110697258 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -2.5199309894899704e+102`

`if -2.5199309894899704e+102 < c < 1.8011763110697258e+137`

`if 1.8011763110697258e+137 < c`

Reproduce