Average Error: 26.6 → 26.9
Time: 9.9s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 2.66647901449877398 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;c \le 1.302160936926946 \cdot 10^{147}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{a \cdot c + b \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le 2.66647901449877398 \cdot 10^{55}:\\
\;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;c \le 1.302160936926946 \cdot 10^{147}:\\
\;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{a \cdot c + b \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r168466 = a;
        double r168467 = c;
        double r168468 = r168466 * r168467;
        double r168469 = b;
        double r168470 = d;
        double r168471 = r168469 * r168470;
        double r168472 = r168468 + r168471;
        double r168473 = r168467 * r168467;
        double r168474 = r168470 * r168470;
        double r168475 = r168473 + r168474;
        double r168476 = r168472 / r168475;
        return r168476;
}
↓
double f(double a, double b, double c, double d) {
        double r168477 = c;
        double r168478 = 2.666479014498774e+55;
        bool r168479 = r168477 <= r168478;
        double r168480 = a;
        double r168481 = r168480 * r168477;
        double r168482 = b;
        double r168483 = d;
        double r168484 = r168482 * r168483;
        double r168485 = r168481 + r168484;
        double r168486 = r168477 * r168477;
        double r168487 = r168483 * r168483;
        double r168488 = r168486 + r168487;
        double r168489 = sqrt(r168488);
        double r168490 = r168485 / r168489;
        double r168491 = r168490 / r168489;
        double r168492 = 1.3021609369269456e+147;
        bool r168493 = r168477 <= r168492;
        double r168494 = r168480 / r168489;
        double r168495 = 1.0;
        double r168496 = r168488 / r168485;
        double r168497 = r168495 / r168496;
        double r168498 = r168493 ? r168494 : r168497;
        double r168499 = r168479 ? r168491 : r168498;
        return r168499;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.6
Target0.4
Herbie26.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if c < 2.666479014498774e+55

    1. Initial program 23.8

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt23.8

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*23.7

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    if 2.666479014498774e+55 < c < 1.3021609369269456e+147

    1. Initial program 22.1

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt22.1

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*22.0

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around inf 26.3

      \[\leadsto \frac{\color{blue}{a}}{\sqrt{c \cdot c + d \cdot d}}\]

    if 1.3021609369269456e+147 < c

    1. Initial program 44.8

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity44.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d}\]
    4. Using strategy rm

    5. Applied clear-num44.8

      \[\leadsto \color{blue}{\frac{1}{\frac{c \cdot c + d \cdot d}{1 \cdot \left(a \cdot c + b \cdot d\right)}}}\]

    6. Simplified44.8

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot c + d \cdot d}{a \cdot c + b \cdot d}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 2.66647901449877398 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;c \le 1.302160936926946 \cdot 10^{147}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{a \cdot c + b \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))