Complex division, real part

Average Error: 25.8 → 14.6

Time: 14.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 9.936266958075772 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r3106301 = a;
        double r3106302 = c;
        double r3106303 = r3106301 * r3106302;
        double r3106304 = b;
        double r3106305 = d;
        double r3106306 = r3106304 * r3106305;
        double r3106307 = r3106303 + r3106306;
        double r3106308 = r3106302 * r3106302;
        double r3106309 = r3106305 * r3106305;
        double r3106310 = r3106308 + r3106309;
        double r3106311 = r3106307 / r3106310;
        return r3106311;
}

↓

double f(double a, double b, double c, double d) {
        double r3106312 = d;
        double r3106313 = 9.936266958075772e+94;
        bool r3106314 = r3106312 <= r3106313;
        double r3106315 = 1.0;
        double r3106316 = c;
        double r3106317 = hypot(r3106312, r3106316);
        double r3106318 = a;
        double r3106319 = b;
        double r3106320 = r3106319 * r3106312;
        double r3106321 = fma(r3106318, r3106316, r3106320);
        double r3106322 = r3106317 / r3106321;
        double r3106323 = r3106315 / r3106322;
        double r3106324 = r3106323 / r3106317;
        double r3106325 = r3106319 / r3106317;
        double r3106326 = r3106314 ? r3106324 : r3106325;
        return r3106326;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.8
Target	0.4
Herbie	14.6

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

Derivation

1. Split input into 2 regimes

2. if d < 9.936266958075772e+94

   1. Initial program 22.9

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

   2. Simplified 22.9

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

   4. Applied add-sqr-sqrt 22.9

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

   5. Applied associate-/r* 22.8

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

   7. Applied fma-udef 22.8

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

   8. Applied hypot-def 22.8

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

   10. Applied fma-udef 22.8

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

   11. Applied hypot-def 14.0

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

   13. Applied clear-num 14.1

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

   if 9.936266958075772e+94 < d

   1. Initial program 38.8

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

   2. Simplified 38.8

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

   4. Applied add-sqr-sqrt 38.8

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

   5. Applied associate-/r* 38.7

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

   7. Applied fma-udef 38.7

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

   8. Applied hypot-def 38.7

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

   10. Applied fma-udef 38.7

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

   11. Applied hypot-def 25.9

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

   12. Taylor expanded around 0 16.7

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

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 9.936266958075772 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))))