Average Error: 25.9 → 12.8
Time: 12.4s
Precision: binary64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{d}{c}, b, a\right)\\ \mathbf{if}\;c \leq -6.842778186148658 \cdot 10^{-25}:\\ \;\;\;\;\frac{-t_0}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.2529092715176455 \cdot 10^{-205}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 4.291969600457141 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(\frac{d}{c}, b, a\right)\\
\mathbf{if}\;c \leq -6.842778186148658 \cdot 10^{-25}:\\
\;\;\;\;\frac{-t_0}{\mathsf{hypot}\left(d, c\right)}\\

\mathbf{elif}\;c \leq 2.2529092715176455 \cdot 10^{-205}:\\
\;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\

\mathbf{elif}\;c \leq 4.291969600457141 \cdot 10^{+42}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0}{\mathsf{hypot}\left(d, 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 (/ d c) b a)))
   (if (<= c -6.842778186148658e-25)
     (/ (- t_0) (hypot d c))
     (if (<= c 2.2529092715176455e-205)
       (fma (/ a d) (/ c d) (/ b d))
       (if (<= c 4.291969600457141e+42)
         (/ (/ (fma a c (* d b)) (hypot d c)) (hypot d c))
         (/ t_0 (hypot d 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((d / c), b, a);
	double tmp;
	if (c <= -6.842778186148658e-25) {
		tmp = -t_0 / hypot(d, c);
	} else if (c <= 2.2529092715176455e-205) {
		tmp = fma((a / d), (c / d), (b / d));
	} else if (c <= 4.291969600457141e+42) {
		tmp = (fma(a, c, (d * b)) / hypot(d, c)) / hypot(d, c);
	} else {
		tmp = t_0 / hypot(d, c);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.9
Target0.4
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if c < -6.8427781861486576e-25

    1. Initial program 30.1

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

    2. Simplified30.0

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

    3. Applied add-sqr-sqrt_binary6430.1

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

    4. Applied *-un-lft-identity_binary6430.1

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

    5. Applied times-frac_binary6430.1

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

    6. Simplified30.1

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

    7. Simplified20.9

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

    8. Applied associate-*l/_binary6420.8

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

    9. Simplified20.8

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

    10. Taylor expanded in c around -inf 18.3

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

    11. Simplified16.2

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

    if -6.8427781861486576e-25 < c < 2.2529092715176455e-205

    1. Initial program 21.1

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

    2. Simplified21.1

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

    3. Applied add-sqr-sqrt_binary6421.1

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

    4. Applied *-un-lft-identity_binary6421.1

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

    5. Applied times-frac_binary6421.1

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

    6. Simplified21.1

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

    7. Simplified12.2

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

    8. Taylor expanded in d around inf 13.0

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

    9. Simplified11.2

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

    if 2.2529092715176455e-205 < c < 4.2919696004571409e42

    1. Initial program 16.8

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

    2. Simplified16.8

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

    3. Applied add-sqr-sqrt_binary6416.8

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

    4. Applied *-un-lft-identity_binary6416.8

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

    5. Applied times-frac_binary6416.8

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

    6. Simplified16.8

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

    7. Simplified10.5

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

    8. Applied associate-*l/_binary6410.4

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

    9. Simplified10.4

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

    10. Applied *-un-lft-identity_binary6410.4

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

    if 4.2919696004571409e42 < c

    1. Initial program 34.9

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

    2. Simplified34.9

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

    3. Applied add-sqr-sqrt_binary6434.9

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

    4. Applied *-un-lft-identity_binary6434.9

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

    5. Applied times-frac_binary6434.9

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

    6. Simplified34.9

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

    7. Simplified23.9

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

    8. Applied associate-*l/_binary6423.8

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

    9. Simplified23.8

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

    10. Taylor expanded in c around inf 15.7

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

    11. Simplified12.9

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.842778186148658 \cdot 10^{-25}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.2529092715176455 \cdot 10^{-205}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 4.291969600457141 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021225 
(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))))