Average Error: 26.1 → 7.2
Time: 7.1s
Precision: binary64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
↓
\[\begin{array}{l} \mathbf{if}\;d \leq -1.3391945132269083 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, a, b\right) \cdot \frac{-1}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq 2.6948582591098956 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}, a, \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \end{array} \]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \leq -1.3391945132269083 \cdot 10^{+119}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{d}, a, b\right) \cdot \frac{-1}{\mathsf{hypot}\left(d, c\right)}\\

\mathbf{elif}\;d \leq 2.6948582591098956 \cdot 10^{+146}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}, a, \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\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
 (if (<= d -1.3391945132269083e+119)
   (* (fma (/ c d) a b) (/ -1.0 (hypot d c)))
   (if (<= d 2.6948582591098956e+146)
     (fma
      (* (/ 1.0 (hypot c d)) (/ c (hypot c d)))
      a
      (/ (* d b) (fma d d (* c c))))
     (fma (/ a d) (/ c d) (/ b d)))))
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 tmp;
	if (d <= -1.3391945132269083e+119) {
		tmp = fma((c / d), a, b) * (-1.0 / hypot(d, c));
	} else if (d <= 2.6948582591098956e+146) {
		tmp = fma(((1.0 / hypot(c, d)) * (c / hypot(c, d))), a, ((d * b) / fma(d, d, (c * c))));
	} else {
		tmp = fma((a / d), (c / d), (b / d));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.1
Target0.4
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if d < -1.33919451322690825e119

    1. Initial program 41.2

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

    2. Simplified41.2

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

    3. Applied add-sqr-sqrt_binary6441.2

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

    4. Applied *-un-lft-identity_binary6441.2

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

    5. Applied times-frac_binary6441.2

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

    6. Simplified41.2

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

    7. Simplified27.3

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

    8. Taylor expanded in d around -inf 12.6

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

    9. Simplified8.0

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

    if -1.33919451322690825e119 < d < 2.69485825910989561e146

    1. Initial program 18.7

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

    2. Simplified18.7

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

    3. Applied add-sqr-sqrt_binary6418.7

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

    4. Applied *-un-lft-identity_binary6418.7

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

    5. Applied times-frac_binary6418.8

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

    6. Simplified18.8

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

    7. Simplified12.1

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

    8. Taylor expanded in b around 0 18.7

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

    9. Simplified16.6

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

    10. Applied add-sqr-sqrt_binary6416.6

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

    11. Applied *-un-lft-identity_binary6416.6

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

    12. Applied times-frac_binary6416.6

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

    13. Simplified16.6

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

    14. Simplified6.9

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

    if 2.69485825910989561e146 < d

    1. Initial program 44.6

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

    2. Simplified44.6

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

    3. Applied add-sqr-sqrt_binary6444.6

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

    4. Applied *-un-lft-identity_binary6444.6

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

    5. Applied times-frac_binary6444.6

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

    6. Simplified44.6

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

    7. Simplified28.8

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

    8. Taylor expanded in d around inf 14.7

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

    9. Simplified7.5

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3391945132269083 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, a, b\right) \cdot \frac{-1}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq 2.6948582591098956 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}, a, \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))