Average Error: 26.4 → 10.4
Time: 4.5s
Precision: binary64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
↓
\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -9.705756685731516 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, b, a\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -4.084260136662665 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.5110647824345223 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 3.408225542871235 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)\\ \end{array} \]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
t_0 := \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\
\mathbf{if}\;c \leq -9.705756685731516 \cdot 10^{+151}:\\
\;\;\;\;\mathsf{fma}\left(\frac{d}{c}, b, a\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\

\mathbf{elif}\;c \leq -4.084260136662665 \cdot 10^{-238}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;c \leq 4.5110647824345223 \cdot 10^{-212}:\\
\;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\

\mathbf{elif}\;c \leq 3.408225542871235 \cdot 10^{+180}:\\
\;\;\;\;t_0\\

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


\end{array}

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (/ (fma a c (* d b)) (hypot c d)) (hypot c d))))
   (if (<= c -9.705756685731516e+151)
     (* (fma (/ d c) b a) (/ -1.0 (hypot c d)))
     (if (<= c -4.084260136662665e-238)
       t_0
       (if (<= c 4.5110647824345223e-212)
         (fma (/ a d) (/ c d) (/ b d))
         (if (<= c 3.408225542871235e+180)
           t_0
           (fma (/ b c) (/ d c) (/ a c))))))))
double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}
↓
double code(double a, double b, double c, double d) {
	double t_0 = (fma(a, c, (d * b)) / hypot(c, d)) / hypot(c, d);
	double tmp;
	if (c <= -9.705756685731516e+151) {
		tmp = fma((d / c), b, a) * (-1.0 / hypot(c, d));
	} else if (c <= -4.084260136662665e-238) {
		tmp = t_0;
	} else if (c <= 4.5110647824345223e-212) {
		tmp = fma((a / d), (c / d), (b / d));
	} else if (c <= 3.408225542871235e+180) {
		tmp = t_0;
	} else {
		tmp = fma((b / c), (d / c), (a / c));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.4
Target0.5
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if c < -9.7057566857315155e151

    1. Initial program 44.9

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

    2. Simplified44.9

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

    3. Applied egg-rr29.5

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

    4. Taylor expanded in c around -inf 11.3

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

    5. Simplified7.4

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

    if -9.7057566857315155e151 < c < -4.08426013666266484e-238 or 4.51106478243452233e-212 < c < 3.408225542871235e180

    1. Initial program 19.4

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

    2. Simplified19.4

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

    3. Applied egg-rr12.5

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

    4. Applied egg-rr12.4

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

    if -4.08426013666266484e-238 < c < 4.51106478243452233e-212

    1. Initial program 23.3

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

    2. Simplified23.3

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

    3. Applied egg-rr12.7

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

    4. Taylor expanded in c around 0 9.2

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]

    5. Simplified6.2

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

    if 3.408225542871235e180 < c

    1. Initial program 45.9

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

    2. Simplified45.9

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

    3. Applied egg-rr32.3

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

    4. Taylor expanded in c around inf 14.8

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]

    5. Simplified6.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.705756685731516 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, b, a\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -4.084260136662665 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 4.5110647824345223 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 3.408225542871235 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))