Average Error: 26.0 → 25.8
Time: 13.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\
\;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r91394 = a;
        double r91395 = c;
        double r91396 = r91394 * r91395;
        double r91397 = b;
        double r91398 = d;
        double r91399 = r91397 * r91398;
        double r91400 = r91396 + r91399;
        double r91401 = r91395 * r91395;
        double r91402 = r91398 * r91398;
        double r91403 = r91401 + r91402;
        double r91404 = r91400 / r91403;
        return r91404;
}
↓
double f(double a, double b, double c, double d) {
        double r91405 = c;
        double r91406 = 1.7284643974157362e+95;
        bool r91407 = r91405 <= r91406;
        double r91408 = a;
        double r91409 = r91408 * r91405;
        double r91410 = d;
        double r91411 = b;
        double r91412 = r91410 * r91411;
        double r91413 = r91409 + r91412;
        double r91414 = r91405 * r91405;
        double r91415 = r91410 * r91410;
        double r91416 = r91414 + r91415;
        double r91417 = sqrt(r91416);
        double r91418 = r91413 / r91417;
        double r91419 = r91418 / r91417;
        double r91420 = r91408 / r91417;
        double r91421 = r91407 ? r91419 : r91420;
        return r91421;
}

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

Original26.0
Target0.4
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < 1.7284643974157362e+95

    1. Initial program 23.0

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

    3. Applied add-sqr-sqrt23.0

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

    4. Applied associate-/r*22.9

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

    5. Simplified22.9

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

    if 1.7284643974157362e+95 < c

    1. Initial program 39.0

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

    3. Applied add-sqr-sqrt39.0

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

    4. Applied associate-/r*39.0

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

    5. Simplified39.0

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

    6. Taylor expanded around inf 38.3

      \[\leadsto \frac{\color{blue}{a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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))))