Average Error: 25.8 → 26.2
Time: 15.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 28148546215012511376658377374508777472:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le 28148546215012511376658377374508777472:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r4366130 = a;
        double r4366131 = c;
        double r4366132 = r4366130 * r4366131;
        double r4366133 = b;
        double r4366134 = d;
        double r4366135 = r4366133 * r4366134;
        double r4366136 = r4366132 + r4366135;
        double r4366137 = r4366131 * r4366131;
        double r4366138 = r4366134 * r4366134;
        double r4366139 = r4366137 + r4366138;
        double r4366140 = r4366136 / r4366139;
        return r4366140;
}
↓
double f(double a, double b, double c, double d) {
        double r4366141 = d;
        double r4366142 = 2.814854621501251e+37;
        bool r4366143 = r4366141 <= r4366142;
        double r4366144 = a;
        double r4366145 = c;
        double r4366146 = b;
        double r4366147 = r4366141 * r4366146;
        double r4366148 = fma(r4366144, r4366145, r4366147);
        double r4366149 = r4366145 * r4366145;
        double r4366150 = fma(r4366141, r4366141, r4366149);
        double r4366151 = sqrt(r4366150);
        double r4366152 = r4366148 / r4366151;
        double r4366153 = r4366152 / r4366151;
        double r4366154 = r4366146 / r4366151;
        double r4366155 = r4366143 ? r4366153 : r4366154;
        return r4366155;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.8
Target0.4
Herbie26.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < 2.814854621501251e+37

    1. Initial program 22.8

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

    2. Simplified22.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt22.8

      \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}\]

    5. Applied *-un-lft-identity22.8

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]

    6. Applied times-frac22.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}\]
    7. Using strategy rm

    8. Applied associate-*l/22.7

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}\]

    if 2.814854621501251e+37 < d

    1. Initial program 35.3

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

    2. Simplified35.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt35.3

      \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}\]

    5. Applied associate-/r*35.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}\]

    6. Taylor expanded around 0 37.0

      \[\leadsto \frac{\color{blue}{b}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification26.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 28148546215012511376658377374508777472:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))