Average Error: 26.0 → 24.7
Time: 14.6s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 1.444073714691111652490143880583541065297 \cdot 10^{307}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} = -\infty:\\
\;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 1.444073714691111652490143880583541065297 \cdot 10^{307}:\\
\;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\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 r6309225 = a;
        double r6309226 = c;
        double r6309227 = r6309225 * r6309226;
        double r6309228 = b;
        double r6309229 = d;
        double r6309230 = r6309228 * r6309229;
        double r6309231 = r6309227 + r6309230;
        double r6309232 = r6309226 * r6309226;
        double r6309233 = r6309229 * r6309229;
        double r6309234 = r6309232 + r6309233;
        double r6309235 = r6309231 / r6309234;
        return r6309235;
}
↓
double f(double a, double b, double c, double d) {
        double r6309236 = b;
        double r6309237 = d;
        double r6309238 = r6309236 * r6309237;
        double r6309239 = a;
        double r6309240 = c;
        double r6309241 = r6309239 * r6309240;
        double r6309242 = r6309238 + r6309241;
        double r6309243 = r6309240 * r6309240;
        double r6309244 = r6309237 * r6309237;
        double r6309245 = r6309243 + r6309244;
        double r6309246 = r6309242 / r6309245;
        double r6309247 = -inf.0;
        bool r6309248 = r6309246 <= r6309247;
        double r6309249 = -r6309239;
        double r6309250 = sqrt(r6309245);
        double r6309251 = r6309249 / r6309250;
        double r6309252 = 1.4440737146911117e+307;
        bool r6309253 = r6309246 <= r6309252;
        double r6309254 = r6309242 / r6309250;
        double r6309255 = r6309254 / r6309250;
        double r6309256 = r6309253 ? r6309255 : r6309251;
        double r6309257 = r6309248 ? r6309251 : r6309256;
        return r6309257;
}

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
Herbie24.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < -inf.0 or 1.4440737146911117e+307 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 63.9

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

    3. Applied add-sqr-sqrt63.9

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

    4. Applied associate-/r*63.9

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

    5. Taylor expanded around -inf 59.5

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]

    6. Simplified59.5

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

    if -inf.0 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 1.4440737146911117e+307

    1. Initial program 11.3

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

    3. Applied add-sqr-sqrt11.3

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

    4. Applied associate-/r*11.2

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

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 1.444073714691111652490143880583541065297 \cdot 10^{307}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (a b c d)
  :name "Complex division, real part"

  :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))))