\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.1649944572964785 \cdot 10^{+191}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 6.443797861366497 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

double f(double a, double b, double c, double d) {
        double r15462325 = a;
        double r15462326 = c;
        double r15462327 = r15462325 * r15462326;
        double r15462328 = b;
        double r15462329 = d;
        double r15462330 = r15462328 * r15462329;
        double r15462331 = r15462327 + r15462330;
        double r15462332 = r15462326 * r15462326;
        double r15462333 = r15462329 * r15462329;
        double r15462334 = r15462332 + r15462333;
        double r15462335 = r15462331 / r15462334;
        return r15462335;
}

↓

double f(double a, double b, double c, double d) {
        double r15462336 = c;
        double r15462337 = -1.1649944572964785e+191;
        bool r15462338 = r15462336 <= r15462337;
        double r15462339 = a;
        double r15462340 = -r15462339;
        double r15462341 = d;
        double r15462342 = hypot(r15462341, r15462336);
        double r15462343 = r15462340 / r15462342;
        double r15462344 = 6.443797861366497e+143;
        bool r15462345 = r15462336 <= r15462344;
        double r15462346 = b;
        double r15462347 = r15462341 * r15462346;
        double r15462348 = fma(r15462339, r15462336, r15462347);
        double r15462349 = r15462348 / r15462342;
        double r15462350 = r15462349 / r15462342;
        double r15462351 = r15462339 / r15462342;
        double r15462352 = r15462345 ? r15462350 : r15462351;
        double r15462353 = r15462338 ? r15462343 : r15462352;
        return r15462353;
}

\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}

↓

\begin{array}{l}
\mathbf{if}\;c \le -1.1649944572964785 \cdot 10^{+191}:\\
\;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\

\mathbf{elif}\;c \le 6.443797861366497 \cdot 10^{+143}:\\
\;\;\;\;\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}

Original	25.8
Target	0.4
Herbie	12.5

Derivation

Split input into 3 regimes
if c < -1.1649944572964785e+191
1. Initial program 43.0
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified43.0
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.0
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*43.0
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef43.0
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def43.0
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied fma-udef43.0
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{d^2 + c^2}^*}\]
11. Applied hypot-def29.7
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around -inf 10.9
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
13. Simplified10.9
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -1.1649944572964785e+191 < c < 6.443797861366497e+143
1. Initial program 20.5
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified20.5
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt20.5
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*20.4
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef20.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def20.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied fma-udef20.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{d^2 + c^2}^*}\]
11. Applied hypot-def12.4
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 6.443797861366497e+143 < c
1. Initial program 43.2
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Simplified43.2
  \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt43.2
  \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
5. Applied associate-/r*43.2
  \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
6. Using strategy rm
7. Applied fma-udef43.2
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
8. Applied hypot-def43.2
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
9. Using strategy rm
10. Applied fma-udef43.2
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{d^2 + c^2}^*}\]
11. Applied hypot-def26.9
  \[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
12. Taylor expanded around inf 14.7
  \[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.5
\[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.1649944572964785 \cdot 10^{+191}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 6.443797861366497 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Target

Derivation

`if c < -1.1649944572964785e+191`

`if -1.1649944572964785e+191 < c < 6.443797861366497e+143`

`if 6.443797861366497e+143 < c`

Reproduce